Find the probability that a randomly chosen point in the figure lies in the shaded

VOCABULARY Copy and complete: Either pair of parallel sides of a parallelogram can be called its ? , and the perpendicular distance between these sides is called the ? .

WRITING What are the two formulas you have learned for the area of a rectangle? Explain why these formulas give the same results.

FINDING AREA Find the area of the polygon. 7 4

FINDING AREA Find the area of the polygon. 14 12

FINDING AREA Find the area of the polygon. 15

FINDING AREA Find the area of the polygon. 13 10

FINDING AREA Find the area of the polygon. 30 18

FINDING AREA Find the area of the polygon. 9 15

COMPARING METHODS Show two different ways to calculate the area of parallelogram ABCD. Compare your results. 20 16 8 E D C A

ERROR ANALYSIS Describe and correct the error in finding the area of the parallelogram. A 5 bh 5 (6)(5) 5 30 6 5 4

ERROR ANALYSIS Describe and correct the error in finding the area of the parallelogram. A 5 bh 5 (7)(4) 5 28 4 3

PYTHAGOREAN THEOREM The lengths of the hypotenuse and one leg of a right triangle are given. Find the perimeter and area of the triangle. Hypotenuse: 15 in.; leg: 12 in.

PYTHAGOREAN THEOREM The lengths of the hypotenuse and one leg of a right triangle are given. Find the perimeter and area of the triangle. Hypotenuse: 34 ft; leg: 16 ft

PYTHAGOREAN THEOREM The lengths of the hypotenuse and one leg of a right triangle are given. Find the perimeter and area of the triangle. Hypotenuse: 85 m; leg: 84 m

PYTHAGOREAN THEOREM The lengths of the hypotenuse and one leg of a right triangle are given. Find the perimeter and area of the triangle. Hypotenuse: 29 cm; leg: 20 cm

ALGEBRA Find the value of x. A 5 36 in.2 x 12 in.

ALGEBRA Find the value of x. A 5 276 ft2 12 ft x

ALGEBRA Find the value of x. A 5 476 cm2 17 cm x

ALGEBRA The area of a triangle is 4 square feet. The height of the triangle is half its base. Find the base and the height.

ALGEBRA The area of a parallelogram is 507 square centimeters, and its height is three times its base. Find the base and the height.

OPEN-ENDED MATH A polygon has an area of 80 square meters and a height of 10 meters. Make scale drawings of three different triangles and three different parallelograms that match this description. Label the base and the height.

FINDING AREA Find the area of the shaded polygon. 17 ft 8 ft 5

FINDING AREA Find the area of the shaded polygon. 18 cm 13 cm 9 cm 11 cm

FINDING AREA Find the area of the shaded polygon. 16 m 10 m 11

FINDING AREA Find the area of the shaded polygon. 19 in. 25 in. 1

FINDING AREA Find the area of the shaded polygon. 10 m 26 m 40 m 20 m

FINDING AREA Find the area of the shaded polygon. 8 in. 5 in.

COORDINATE GRAPHING Graph the points and connect them to form a polygon. Find the area of the polygon. A(3, 3), B(10, 3), C(8, 23), D(1, 23)

COORDINATE GRAPHING Graph the points and connect them to form a polygon. Find the area of the polygon. E(22, 22), F(5, 1), G(3, 22)

MULTIPLE CHOICE What is the area of the parallelogram shown at the right? A 8 ft2 6 in.2 B 1350 in. C 675 in.2 D 9.375 ft2 2 ft 3 in. 4 ft 2 in.

TECHNOLOGY Use geometry drawing software to draw a line l and a line m parallel to l. Then draw nABC so that C is on line l and }AB is on line m. Find the base AB, the height CD, and the area of nABC. Move point C to change the shape of nABC. What do you notice about the base, height, and area of nABC?

USING TRIGONOMETRY In ~ABCD, base AD is 15 and AB is 8. What are the height and area of ~ABCD if mDAB is 208? if mDAB is 508?

ALGEBRA Find the area of a right triangle with side lengths 12 centimeters, 35 centimeters, and 37 centimeters. Then find the length of the altitude drawn to the hypotenuse.

ALGEBRA Find the area of a triangle with side lengths 5 feet, 5 feet, and 8 feet. Then find the lengths of all three altitudes of the triangle.

CHALLENGE The vertices of quadrilateral ABCD are A(2, 22), B(6, 4), C(21, 5), and D(25, 2). Without using the Distance Formula, find the area of ABCD. Show your steps.

SAILING Sails A and B are right triangles. The lengths of the legs of Sail A are 65 feet and 35 feet. The lengths of the legs of Sail B are 29.5 feet and 10.5 feet. Find the area of each sail to the nearest square foot. About how many times as great is the area of Sail A as the area of Sail B? A B

MOWING You can mow 10 square yards of grass in one minute. How long does it take you to mow a triangular plot with height 25 yards and base 24 yards? How long does it take you to mow a rectangular plot with base 24 yards and height 36 yards?

CARPENTRY You are making a table in the shape of a parallelogram to replace an old 24 inch by 15 inch rectangular table. You want the areas of two tables to be equal. The base of the parallelogram is 20 inches. What should the height be?

SHORT RESPONSE A 4 inch square is a square that has a side length of 4 inches. Does a 4 inch square have an area of 4 square inches? If not, what size square does have an area of 4 square inches? Explain.

PAINTING You are earning money by painting a shed. You plan to paint two sides of the shed today. Each of the two sides has the dimensions shown at the right. You can paint 200 square feet per hour, and you charge $20 per hour. How much will you get paid for painting those two sides of the shed? 308 12 ft 6.5 ft

ENVELOPES The pattern below shows how to make an envelope to fit a card that is 17 centimeters by 14 centimeters. What are the dimensions of the rectangle you need to start with? What is the area of the paper that is actually used in the envelope? of the paper that is cut off? 17 cm 14 cm 17 cm 14 cm CM CM CARDFITSHERE CM FOLDLINE CM CM CM

JUSTIFYING THEOREM 11.2 You can use the area formula for a rectangle to justify the area formula for a parallelogram. First draw ~PQRS with base b and height h, as shown. Then draw a segment perpendicular to ] PS through point R. Label point V. a. In the diagram, explain how you know that nPQT > nSRV. b. Explain how you know that the area of PQRS is equal to the area of QRVT. How do you know that Area of PQRS 5 bh? P R P T SV

JUSTIFYING THEOREM 11.3 You can use the area formula for a parallelogram to justify the area formula for a triangle. Start with two congruent triangles with base b and height h. Place and label them as shown. Explain how you know that XYZW is a parallelogram and that Area of nXYW 5 1 }2 bh. Y Z X W

MULTI-STEP PROBLEM You have enough silver to make a pendant with an area of 4 square centimeters. The pendant will be an equilateral triangle. Let s be the side length of the triangle. a. Find the height h of the triangle in terms of s. Then write a formula for the area of the triangle in terms of s. b. Find the side length of the triangle. Round to the nearest centimeter.

EXTENDED RESPONSE The base of a parallelogram is 7 feet and the height is 3 feet. Explain why the perimeter cannot be determined from the given information. Is there a least possible perimeter for the parallelogram? Is there a greatest possible perimeter? Explain.

JUSTIFYING THEOREM 11.1 You can use the diagram to show that the area of a rectangle is the product of its base b and height h. a. Figures MRVU and VSPT are congruent rectangles with base b and height h. Explain why RNSV, UVTQ, and MNPQ are squares. Write expressions in terms of b and h for the areas of the squares. b. Let A be the area of MRVU. Substitute A and the expressions from part (a) into the equation below. Solve to find an expression for A. Area of MNPQ 5 Area of MRVU 1 Area of UVTQ 1 Area of RNSV 1 Area of VSPT M b R N P T h P U V h S b

CHALLENGE An equation of ] AB is y 5 x. An equation of ] AC is y 5 2. Suppose ] BC is placed so that nABC is isosceles with an area of 4 square units. Find two different lines that fit these conditions. Give an equation for each line. Is there another line that could fit this requirement for ] BC ? Explain. y y 5 2 y 5 x A C B

Find the length of the midsegment}MN of the trapezoid. (p. 542) M N 1

Find the length of the midsegment}MN of the trapezoid. (p. 542) 13 27

Find the length of the midsegment}MN of the trapezoid. (p. 542) 46 29

The coordinates of nPQR are P(24, 1), Q(2, 5), and R(1, 24). Graph the image of the triangle after the translation. Use prime notation. (p. 572) (x, y) (x 1 1, y 1 4)

The coordinates of nPQR are P(24, 1), Q(2, 5), and R(1, 24). Graph the image of the triangle after the translation. Use prime notation. (p. 572) (x, y) (x 1 3, y 2 5)

The coordinates of nPQR are P(24, 1), Q(2, 5), and R(1, 24). Graph the image of the triangle after the translation. Use prime notation. (p. 572) (x, y) (x 2 3, y 2 2)

The coordinates of nPQR are P(24, 1), Q(2, 5), and R(1, 24). Graph the image of the triangle after the translation. Use prime notation. (p. 572) (x, y) (x 2 2, y 1 3)

VOCABULARY Copy and complete: The perpendicular distance between the bases of a trapezoid is called the ? of the trapezoid.

WRITING Sketch a kite and its diagonals. Describe what you know about the segments and angles formed by the intersecting diagonals.

FINDING AREA Find the area of the trapezoid. 10 11 8

FINDING AREA Find the area of the trapezoid. 6 6 1

FINDING AREA Find the area of the trapezoid. 5 4.8 7

DRAWING DIAGRAMS The lengths of the bases of a trapezoid are 5.4 centimeters and 10.2 centimeters. The height is 8 centimeters. Draw and label a trapezoid that matches this description. Then find its area.

FINDING AREA Find the area of the rhombus or kite. 60 5

FINDING AREA Find the area of the rhombus or kite. 48 16

FINDING AREA Find the area of the rhombus or kite. 21 18

FINDING AREA Find the area of the rhombus or kite. 10 19

FINDING AREA Find the area of the rhombus or kite. 12 15

FINDING AREA Find the area of the rhombus or kite. 2 4 5

ERROR ANALYSIS Describe and correct the error in finding the area. 14 cm 13 cm 12 cm A 5 1 }2 (13)(14 1 19) 5 214.5 cm2

ERROR ANALYSIS Describe and correct the error in finding the area. 12 cm 5 cm 16 cm A 5 1 }2 (12)(21) 5 126 cm2

MULTIPLE CHOICE One diagonal of a rhombus is three times as long as the other diagonal. The area of the rhombus is 24 square feet. What are the lengths of the diagonals? A 8 ft, 11 ft B 4 ft, 12 ft C 2 ft, 6 ft D 6 ft, 24 ft

ALGEBRA Use the given information to find the value of x. Area 5 108 ft2 14 ft 22 ft x

ALGEBRA Use the given information to find the value of x. Area 5 300 m2 x 10 m 20 m

ALGEBRA Use the given information to find the value of x. Area 5 100 yd2 10 yd x

COORDINATE GEOMETRY Find the area of the figure. y 1 x

COORDINATE GEOMETRY Find the area of the figure. y 4 x 1

COORDINATE GEOMETRY Find the area of the figure. y x 4 1

ALGEBRA Find the lengths of the bases of the trapezoid described. The height is 3 feet. One base is twice as long as the other base. The area is 13.5 square feet.

ALGEBRA Find the lengths of the bases of the trapezoid described. One base is 8 centimeters longer than the other base. The height is 6 centimeters and the area is 54 square centimeters.

FINDING AREA Find the area of the shaded region. 20 16 30

FINDING AREA Find the area of the shaded region. 15 8 9

FINDING AREA Find the area of the shaded region. 20 21 29

FINDING AREA Find the area of the shaded region. 10 5 7

FINDING AREA Find the area of the shaded region. 5 4

FINDING AREA Find the area of the shaded region. 12 7

OPEN-ENDED MATH Draw three examples of trapezoids that match this description: The height of the trapezoid is 3 units and its area is the same as the area of a parallelogram with height 3 units and base 8 units.

VISUALIZING Sketch the figure. Then determine its perimeter and area. The figure is a trapezoid. It has two right angles. The lengths of its bases are 7 and 15. Its height is 6.

VISUALIZING Sketch the figure. Then determine its perimeter and area. The figure is a rhombus. Its side length is 13. The length of one of its diagonals is 24.

CHALLENGE In the diagram shown at the right, ABCD is a parallelogram and BF 5 16. Find the area of ~ABCD. Explain your reasoning. (Hint: Draw auxiliary lines through point A and through point D that are parallel to}EH .) E 8 3 A B

TRUCKS The windshield in a truck is in the shape of a trapezoid. The lengths of the bases of the trapezoid are 70 inches and 79 inches. The height is 35 inches. Find the area of the glass in the windshield.

INTERNET You are creating a kite-shaped logo for your schools website. The diagonals of the logo are 8 millimeters and 5 millimeters long. Find the area of the logo. Draw two different possible shapes for the logo.

DESIGN You are designing a wall hanging that is in the shape of a rhombus. The area of the wall hanging is 432 square inches and the length of one diagonal is 36 inches. Find the length of the other diagonal.

MULTI-STEP PROBLEM As shown, a baseball stadiums playing field is shaped like a pentagon. To find the area of the playing field shown at the right, you can divide the field into two smaller polygons. a. Classify the two polygons. b. Find the area of the playing field in square feet. Then express your answer in square yards. Round to the nearest square foot. 145 ft 179 ft 450 ft 315 ft 322 ft

VISUAL REASONING Follow the steps in parts (a)(c). a. Analyze Copy the table and extend it to include a column for n 5 5. Complete the table for n 5 4 and n 5 5. b. Use Algebra Describe the relationship between the rhombus number n and the area of the rhombus. Then write an algebraic rule for finding the area of the nth rhombus. c. Compare In each rhombus, the length of one diagonal (d1) is 2. What is the length of the other diagonal (d2) for the nth rhombus? Use the formula for the area of a rhombus to write a rule for finding the area of the nth rhombus. Compare this rule with the one you wrote in part (b). Rhombus number, n 12 3 4 Diagram Area, A 24 6 ?

SHORT RESPONSE Look back at the Activity on page 729. Explain how the results for kites in Explore 2 can be used to justify Theorem 11.5, the formula for the area of a rhombus.

PROVING THEOREMS 11.4 AND 11.6 Use the triangle area formula and the triangles in the diagram to write a plan for the proof. Show that the area A of the trapezoid shown is 1}2 h(b1 1 b2). S b1 R h P

PROVING THEOREMS 11.4 AND 11.6 Use the triangle area formula and the triangles in the diagram to write a plan for the proof. Show that the area A of the kite shown is 1}2 d1d2. P R S P

EXTENDED RESPONSE You will explore the effect of moving a diagonal. a. Investigate Draw a kite in which the longer diagonal is horizontal. Suppose this diagonal is fixed and you can slide the vertical diagonal left or right and up or down. You can keep sliding as long as the diagonals continue to intersect. Draw and identify each type of figure you can form. b. Justify Is it possible to form any shapes that are not quadrilaterals? Explain. c. Compare Compare the areas of the different shapes you found in part (b). What do you notice about the areas? Explain. D A C B D A Moved }BD closer to C. Didnt move it up or down. original kite still a kite

CHALLENGE James A. Garfield, the twentieth president of the United States, discovered a proof of the Pythagorean Theorem in 1876. His proof involved the fact that a trapezoid can be formed from two congruent right triangles and an isosceles right triangle. Use the diagram to show that a2 1 b2 5 c2 . b b c c

Solve for the indicated variable. Write a reason for each step. (p. 105) d 5 rt; solve for t

Solve for the indicated variable. Write a reason for each step. (p. 105) A 5 1 }2 bh; solve for h

Solve for the indicated variable. Write a reason for each step. (p. 105) P 5 2l 1 2w; solve for w

Find the angle measures of an isosceles triangle if the measure of a base angle is 4 times the measure of the vertex angle. (p. 264)

In the diagram at the right, nPQR , nSTU. The perimeter of nSTU is 81 inches. Find the height h and the perimeter of nPQR. (p. 372) 20 P P R h 30 18 S

VOCABULARY Sketch two similar triangles. Use your sketch to explain what is meant by corresponding side lengths.

WRITING Two regular n-gons are similar. The ratio of their side lengths is is 3 : 4. Do you need to know the value of n to find the ratio of the perimeters or the ratio of the areas of the polygons? Explain.

FINDING RATIOS Copy and complete the table of ratios for similar polygons. Ratio of corresponding side lengths Ratio of perimeters Ratio of areas 6 : 11 ? ? ? 20 : 36 5 ??

FINDING RATIOS Copy and complete the table of ratios for similar polygons. Ratio of corresponding side lengths Ratio of perimeters Ratio of areas 6 : 11 ? ? ? 20 : 36 5 ??

RATIOS AND AREAS Corresponding lengths in similar figures are given. Find the ratios (red to blue) of the perimeters and areas. Find the unknown area. A 5 2 ft 6 ft

RATIOS AND AREAS Corresponding lengths in similar figures are given. Find the ratios (red to blue) of the perimeters and areas. Find the unknown area. 15 cm 20 cm A 5 240 cm2

RATIOS AND AREAS Corresponding lengths in similar figures are given. Find the ratios (red to blue) of the perimeters and areas. Find the unknown area. 7 in. 9 in. A 5 210 in.2

RATIOS AND AREAS Corresponding lengths in similar figures are given. Find the ratios (red to blue) of the perimeters and areas. Find the unknown area. 5 yd 3 yd A 5

FINDING LENGTH RATIOS The ratio of the areas of two similar figures is given. Write the ratio of the lengths of corresponding sides. Ratio of areas 5 49 : 16

FINDING LENGTH RATIOS The ratio of the areas of two similar figures is given. Write the ratio of the lengths of corresponding sides. Ratio of areas 5 16: 121

FINDING LENGTH RATIOS The ratio of the areas of two similar figures is given. Write the ratio of the lengths of corresponding sides. Ratio of areas 5 121 : 144

MULTIPLE CHOICE The area of nLMN is 18 ft2 and the area of nFGH is 24 ft2 . If nLMN , nFGH, what is the ratio of LM to FG? A 3:4 B 9 : 16 C } 3 :2 D 4:3

FINDING SIDE LENGTHS Use the given area to find XY. nDEF , nXYZ 4 cm A 5 7 cm2 D F A 5 28 cm2 Z EX Y

FINDING SIDE LENGTHS Use the given area to find XY. UVWXY , LMNPQ A 5 198 in.2 V U Y N P P L M X

ERROR ANALYSIS In the diagram, Rectangles DEFG and WXYZ are similar. The ratio of the area of DEFG to the area of WXYZ is 1 : 4. Describe and correct the error in finding ZY. ZY 5 4(12) 5 48 G FZ Y 12 W X D

REGULAR PENTAGONS Regular pentagon QRSTU has a side length of 12 centimeters and an area of about 248 square centimeters. Regular pentagon VWXYZ has a perimeter of 140 centimeters. Find its area.

RHOMBUSES Rhombuses MNPQ and RSTU are similar. The area of RSTU is 28 square feet. The diagonals of MNPQ are 25 feet long and 14 feet long. Find the area of MNPQ. Then use the ratio of the areas to find the lengths of the diagonals of RSTU.

SHORT RESPONSE You enlarge the same figure three different ways. In each case, the enlarged figure is similar to the original. List the enlargements in order from smallest to largest. Explain. Case 1 The side lengths of the original figure are multiplied by 3. Case 2 The perimeter of the original figure is multiplied by 4. Case 3 The area of the original figure is multiplied by 5.

REASONING In Exercises 19 and 20, copy and complete the statement using always, sometimes, or never. Explain your reasoning. Doubling the side length of a square ? doubles the area.

REASONING In Exercises 19 and 20, copy and complete the statement using always, sometimes, or never. Explain your reasoning. Two similar octagons ? have the same perimeter.

FINDING AREA The sides of nABC are 4.5 feet, 7.5 feet, and 9 feet long. The area is about 17 square feet. Explain how to use the area of nABC to find the area of a nDEF with side lengths 6 feet, 10 feet, and 12 feet.

RECTANGLES Rectangles ABCD and DEFG are similar. The length of ABCD is 24 feet and the perimeter is 84 square feet. The width of DEFG is 3 yards. Find the ratio of the area of ABCD to the area of DEFG.

SIMILAR TRIANGLES Explain why the red and blue triangles are similar. Find the ratio (red to blue) of the areas of the triangles. Show your steps. D L N M 21 m E F A 5 294 m

SIMILAR TRIANGLES Explain why the red and blue triangles are similar. Find the ratio (red to blue) of the areas of the triangles. Show your steps. T Y X W V 3 yd

CHALLENGE In the diagram shown, ABCD is a parallelogram. The ratio of the area of nAGB to the area of nCGE is 9 : 25, CG 5 10, and GE 5 15. a. Find AG, GB, GF, and FE. Show your methods. b. Give two area ratios other than 9 : 25 or 25 : 9 for pairs of similar triangles in the figure. Explain. A F ED C B G

BANNER Two rectangular banners from this years music festival are shown. Organizers of next years festival want to design a new banner that will be similar to the banner whose dimensions are given in the photograph. The length of the longest side of the new banner will be 5 feet. Find the area of the new banner. 1 ft 3

PATIO A new patio will be an irregular hexagon. The patio will have two long parallel sides and an area of 360 square feet. The area of a similar shaped patio is 250 square feet, and its long parallel sides are 12.5 feet apart. What will be the corresponding distance on the new patio?

MULTIPLE CHOICE You need 20 pounds of grass seed to plant grass inside the baseball diamond shown. About how many pounds do you need to plant grass inside the softball diamond? A 6 B 9 C 13 D 20 60 ft softball diamond 90 ft baseball diamond

MULTI-STEP PROBLEM Use graph paper for parts (a) and (b). a. Draw a triangle and label its vertices. Find the area of the triangle. b. Mark and label the midpoints of each side of the triangle. Connect the midpoints to form a smaller triangle. Show that the larger and smaller triangles are similar. Then use the fact that the triangles are similar to find the area of the smaller triangle.

JUSTIFYING THEOREM 11.7 Choose a type of polygon for which you know the area formula. Use algebra and the area formula to prove Theorem 11.7 for that polygon. (Hint: Use the ratio for the corresponding side lengths in two similar polygons to express each dimension in one polygon as a }b times the corresponding dimension in the other polygon.)

MISLEADING GRAPHS A student wants to show that the students in a science class prefer mysteries to science fiction books. Over a two month period, the students in the class read 50 mysteries, but only 25 science fiction books. The student makes a bar graph of these data. Explain why the graph is visually misleading. Show how the student could redraw the bar graph. Books Read Recently 60 50 40 30 20 10 0

OPEN-ENDED MATH The ratio of the areas of two similar polygons is 9 : 6. Draw two polygons that fit this description. Find the ratio of their perimeters. Then write the ratio in simplest radical form.

EXTENDED RESPONSE Use the diagram shown at the right. a. Name as many pairs of similar triangles as you can. Explain your reasoning. b. Find the ratio of the areas for one pair of similar triangles. c. Show two ways to find the length of}DE A B D C E F 9 10

CHALLENGE In the diagram, the solid figure is a cube. Quadrilateral JKNM is on a plane that cuts through the cube, with JL 5 KL. a. Explain how you know that nJKL , nMNP. b. Suppose } JK MN 5 1 }3 . Find the ratio of the area of nJKL to the area of one face of the cube. c. Find the ratio of the area of nJKL to the area of pentagon JKQRS. S J L T P N K P R M

Find the circumference of the circle with the given radius r or diameter d. Use p 3.14. Round your answers to the nearest hundredth. (p. 49) d 5 4 cm

Find the circumference of the circle with the given radius r or diameter d. Use p 3.14. Round your answers to the nearest hundredth. (p. 49) d 5 10 ft

Find the circumference of the circle with the given radius r or diameter d. Use p 3.14. Round your answers to the nearest hundredth. (p. 49) r 5 2.5 yd

Find the circumference of the circle with the given radius r or diameter d. Use p 3.14. Round your answers to the nearest hundredth. (p. 49) 38. r 5 3.1 m

Find the value of x. (p. 295) x 10

Find the value of x. (p. 672) x8 858

Find the value of x. (p. 680) 1808 888 x

In Exercises 1 and 2, refer to the diagram of (P shown. A B P r VOCABULARY Copy and complete the equation: } ?2r 5} mCAB? .

In Exercises 1 and 2, refer to the diagram of (P shown. A B P r WRITING Describe the difference between the arc measure and the arc length ofCAB .

USING CIRCUMFERENCE Use the diagram to find the indicated measure. Find the circumference. 6 in.

USING CIRCUMFERENCE Use the diagram to find the indicated measure. Find the circumference. 17 cm

USING CIRCUMFERENCE Use the diagram to find the indicated measure. Find the radius. C 5 63 ft

FINDING EXACT MEASURES Find the indicated measure. The exact circumference of a circle with diameter 5 inches

FINDING EXACT MEASURES Find the indicated measure. The exact radius of a circle with circumference 28 meters

FINDING CIRCUMFERENCE Find the circumference of the red circle. 14

FINDING CIRCUMFERENCE Find the circumference of the red circle. 3 2

FINDING CIRCUMFERENCE Find the circumference of the red circle. 10

FINDING ARC LENGTHS Find the length ofCAB . A B P 6 m 4

FINDING ARC LENGTHS Find the length ofCAB . 14 cm 120

FINDING ARC LENGTHS Find the length ofCAB . B P 8 ft 458

ERROR ANALYSIS A student says that two arcs from different circles have the same arc length if their central angles have the same measure. Explain the error in the students reasoning.

FINDING MEASURES In (P shown at the right, QPR > RPS. Find the indicated measure. R S P 8 ft 608 mCQRS

FINDING MEASURES In (P shown at the right, QPR > RPS. Find the indicated measure. R S P 8 ft 608 Length ofCQRS

FINDING MEASURES In (P shown at the right, QPR > RPS. Find the indicated measure. R S P 8 ft 608 mCQR

FINDING MEASURES In (P shown at the right, QPR > RPS. Find the indicated measure. R S P 8 ft 608 mCRSQ

FINDING MEASURES In (P shown at the right, QPR > RPS. Find the indicated measure. R S P 8 ft 608 Length ofCQR

FINDING MEASURES In (P shown at the right, QPR > RPS. Find the indicated measure. R S P 8 ft 608 Length ofCRSQ

USING ARC LENGTH Find the indicated measure. mCAB A B 10 8.73

USING ARC LENGTH Find the indicated measure. Circumference of (Q C D 7.5 768

USING ARC LENGTH Find the indicated measure. Radius of (Q L M 38.95 2608

FINDING PERIMETERS Find the perimeter of the shaded region. 13 6

FINDING PERIMETERS Find the perimeter of the shaded region. 6 3 3 6

COORDINATE GEOMETRY The equation of a circle is given. Find the circumference of the circle. Write the circumference in terms of p. x2 1 y 2 5 16

COORDINATE GEOMETRY The equation of a circle is given. Find the circumference of the circle. Write the circumference in terms of p. (x 1 2)2 1 (y 2 3)2 5 9

COORDINATE GEOMETRY The equation of a circle is given. Find the circumference of the circle. Write the circumference in terms of p. x2 1 y 2 5 18

ALGEBRA Solve the formula C 5 2r for r. Solve the formula C 5 d for d. Use the rewritten formulas to find r and d when C 5 26.

FINDING VALUES In the table below,CAB refers to the arc of a circle. Copy and complete the table. Radius ? 2 0.8 4.2 ? 4} 2 mCAB 458 608 ? 1838 908 ? Length ofCAB 4 ? 0.3 ? 3.22 2.86

SHORT RESPONSE SupposeCEF is an arc on a circle with radius r. Let x8 be the measure ofCEF . Describe the effect on the length ofCEF if you (a) double the radius of the circle, and (b) double the measure ofCEF .

MULTIPLE CHOICE In the diagram, WY }and }XZ are diameters of (T, and WY 5 XZ 5 6. If mCXY 5 1408, what is the length ofCYZ ? A 2 }3 B 4 }3 C 6 D 4 X W Z T Y

CHALLENGE Find the circumference of a circle inscribed in a rhombus with diagonals that are 12 centimeters and 16 centimeters long. Explain.

FINDING CIRCUMFERENCE In the diagram, the measure of the shaded red angle is 308. The arc length a is 2. Explain how to find the circumference of the blue circle without finding the radius of either the red or the blue circles. 2r r a

TREES A group of students wants to find the diameter of the trunk of a young sequoia tree. The students wrap a rope around the tree trunk, then measure the length of rope needed to wrap one time around the trunk. This length is 21 feet 8 inches. Explain how they can use this length to estimate the diameter of the tree trunk to the nearest half foot.

NSCRIBED SQUARE A square with side length 6 units is inscribed in a circle so that all four vertices are on the circle. Draw a sketch to represent this problem. Find the circumference of the circle.

MEASURING WHEEL As shown, a measuring wheel is used to calculate the length of a path. The diameter of the wheel is 8 inches. The wheel rotates 87 times along the length of the path. About how long is the path?

EXTENDED RESPONSE A motorized scooter has a chain drive. The chain goes around the front and rear sprockets. a. About how long is the chain? Explain. b. Each sprocket has teeth that grip the chain. There are 76 teeth on the larger sprocket, and 15 teeth on the smaller sprocket. About how many teeth are gripping the chain at any given time? Explain. 6 in. 1 8 1 in. 7 16 6

SCIENCE Over 2000 years ago, the Greek scholar Eratosthenes estimated Earths circumference by assuming that the Suns rays are parallel. He chose a day when the Sun shone straight down into a well in the city of Syene. At noon, he measured the angle the Suns rays made with a vertical stick in the city of Alexandria. Eratosthenes assumed that the distance from Syene to Alexandria was equal to about 575 miles. Find m 1. Then estimate Earths circumference. Alexandra Syene stick center of Earth Not drawn to scale sunlight sunlight m

CHALLENGE Suppose}AB is divided into four congruent segments, and semicircles with radius r are drawn. What is the sum of the four arc lengths if the radius of each arc is r? A B r A r B A r B

CHALLENGE Suppose}AB is divided into four congruent segments, and semicircles with radius r are drawn. Suppose that }AB is divided into n congruent segments and that semicircles are drawn, as shown. What will the sum of the arc lengths be for 8 segments? for 16 segments? for n segments? Explain your thinking. B r A r B A r B

Find the area of a circle with radius r. Round to the nearest hundredth. (p. 49) r 5 6 cm

Find the area of a circle with radius r. Round to the nearest hundredth. (p. 49) r 5 4.2 in.

Find the area of a circle with radius r. Round to the nearest hundredth. (p. 49) r 5 83 }4 mi

Find the area of a circle with radius r. Round to the nearest hundredth. (p. 49) r 5 13 }8 in.

Find the value of x. (p. 689) 4 8 5 x

Find the value of x. (p. 689) 4 11 5

Find the value of x. (p. 689) 24 8 x

VOCABULARY Copy and complete: A ? of a circle is the region bounded by two radii of the circle and their intercepted arc.

WRITING Suppose you double the arc measure of a sector in a given circle. Will the area of the sector also be doubled? Explain.

FINDING AREA Find the exact area of a circle with the given radius r or diameter d. Then find the area to the nearest hundredth. r 5 5 in.

FINDING AREA Find the exact area of a circle with the given radius r or diameter d. Then find the area to the nearest hundredth. d 5 16 ft

FINDING AREA Find the exact area of a circle with the given radius r or diameter d. Then find the area to the nearest hundredth. d 5 23 cm

FINDING AREA Find the exact area of a circle with the given radius r or diameter d. Then find the area to the nearest hundredth. r 5 1.5 km

USING AREA In Exercises 79, find the indicated measure. The area of a circle is 154 square meters. Find the radius.

USING AREA In Exercises 79, find the indicated measure. The area of a circle is 380 square inches. Find the radius.

USING AREA In Exercises 79, find the indicated measure. The area of a circle is 676 square centimeters. Find the diameter.

ERROR ANALYSIS In the diagram at the right, the area of (Z is 48 square feet. A student writes a proportion to find the area of sector XZY. Describe and correct the error in writing the proportion. Then find the area of sector XZY Let n be the area of sector XZY. } n 3608 5 } 48 2858 X Y W

FINDING AREA OF SECTORS Find the areas of the sectors formed by DFE. E D F G 608 10 in.

FINDING AREA OF SECTORS Find the areas of the sectors formed by DFE. E D F G 2568 14 cm

FINDING AREA OF SECTORS Find the areas of the sectors formed by DFE. D E F G 1378 28 m

USING AREA OF A SECTOR Use the diagram to find the indicated measure. Find the area of (M. L 1658 A 5 38.51 m2

USING AREA OF A SECTOR Use the diagram to find the indicated measure. Find the area of (M. K J L 508 M A 5 56.87 cm2

USING AREA OF A SECTOR Use the diagram to find the indicated measure. Find the radius of (M. J L K M A 5 12.36 m2 898

FINDING AREA Find the area of the shaded region. 6 m 6 m 6 m

FINDING AREA Find the area of the shaded region. 8 in. 16 in. 2

MULTIPLE CHOICE The diagram shows the shape of a putting green at a miniature golf course. One part of the green is a sector of a circle. To the nearest square foot, what is the area of the putting green? A 46 ft2 B 49 ft2 C 56 ft2 D 75 ft2 3.5 ft 7 ft 3.5 ft 3.5 ft

FINDING MEASURES The area of (M is 260.67 square inches. The area of sector KML is 42 square inches. Find the indicated measure. L K M N Radius of (M

FINDING MEASURES The area of (M is 260.67 square inches. The area of sector KML is 42 square inches. Find the indicated measure. L K M N Circumference of (M

FINDING MEASURES The area of (M is 260.67 square inches. The area of sector KML is 42 square inches. Find the indicated measure. L K M N mCKL

FINDING MEASURES The area of (M is 260.67 square inches. The area of sector KML is 42 square inches. Find the indicated measure. L K M N Perimeter of blue region

FINDING MEASURES The area of (M is 260.67 square inches. The area of sector KML is 42 square inches. Find the indicated measure. L K M N Length ofCKL

FINDING MEASURES The area of (M is 260.67 square inches. The area of sector KML is 42 square inches. Find the indicated measure. L K M N Perimeter of red region

FINDING AREA Find the area of the shaded region. 5 in.

FINDING AREA Find the area of the shaded region. 1098 5.2 ft

FINDING AREA Find the area of the shaded region. 20 in. 20 in.

FINDING AREA Find the area of the shaded region. 17 cm 1808

FINDING AREA Find the area of the shaded region. 2 ft

FINDING AREA Find the area of the shaded region. 3 m 4 m

TANGENT CIRCLES In the diagram at the right, (Q and (P are tangent, and P lies on (Q. The measure ofCRS is 1088. Find the area of the red region, the area of the blue region, and the area of the yellow region. Leave your answers in terms of . R S P P

SIMILARITY Look back at the Perimeters of Similar Polygons Theorem on page 374 and the Areas of Similar Polygons Theorem on page 737. How would you rewrite these theorems to apply to circles? Explain.

ERROR ANALYSIS The ratio of the lengths of two arcs in a circle is 2 : 1. A student claims that the ratio of the areas of the sectors bounded by these arcs is 4 : 1, because 1 2 }12 2 5 4 }1 . Describe and correct the error.

DRAWING A DIAGRAM A square is inscribed in a circle. The same square is also circumscribed about a smaller circle. Draw a diagram. Find the ratio of the area of the large circle to the area of the small circle.

CHALLENGE In the diagram at the right,CFG andCEH are arcs of concentric circles, and }EF and GH}lie on radii of the larger circle. Find the area of the shaded region. F G 10 m 30 m 8 m 8 m

METEOROLOGY The eye of a hurricane is a relatively calm circular region in the center of the storm. The diameter of the eye is typically about 20 miles. If the eye of a hurricane is 20 miles in diameter, what is the area of the land that is underneath the eye?

WALKING The area of a circular pond is about 138,656 square feet. You are going to walk around the entire edge of the pond. About how far will you walk? Give your answer to the nearest foot.

CIRCLE GRAPH The table shows how students get to school. a. Explain why a circle graph is appropriate for the data. b. You will represent each method by a sector of a circle graph. Find the central angle to use for each sector. Then use a protractor and a compass to construct the graph. Use a radius of 2 inches. c. Find the area of each sector in your graph. Method % of Students Bus 65% Walk 25% Other 10%

SHORT RESPONSE It takes about 1 }4 cup of dough to make a tortilla with a 6 inch diameter. How much dough does it take to make a tortilla with a 12 inch diameter? Explain your reasoning.

HIGHWAY SIGNS A new typeface has been designed to make highway signs more readable. One change was to redesign the form of the letters to increase the space inside letters. a. Estimate the interior area for the old and the new a. Then find the percent increase in interior area. b. Do you think the change in interior area is just a result of a change in height and width of the letter a? Explain.

EXTENDED RESPONSE A circular pizza with a 12 inch diameter is enough for you and 2 friends. You want to buy pizza for yourself and 7 friends. A 10 inch diameter pizza with one topping costs $6.99 and a 14 inch diameter pizza with one topping costs $12.99. How many 10 inch and 14 inch pizzas should you buy in each situation below? Explain. a. You want to spend as little money as possible. b. You want to have three pizzas, each with a different topping. c. You want to have as much of the thick outer crust as possible.

JUSTIFYING THEOREM 11.9 You can follow the steps below to justify the formula for the area of a circle with radius r. Divide a circle into 16 congruent sectors. Cut out the sectors. Rearrange the 16 sectors to form a shape resembling a parallelogram. a. Write expressions in terms of r for the approximate height and base of the parallelogram. Then write an expression for its area. b. Explain how your answers to part (a) justify Theorem 11.9.

CHALLENGE Semicircles with diameters equal to the three sides of a right triangle are drawn, as shown. Prove that the sum of the area of the two shaded crescents equals the area of the triangle.

Triangle DEG is isosceles with altitude}DF . Find the given measurement. Explain your reasoning. (p. 319) E D F G 198 3 cm m DFG

Triangle DEG is isosceles with altitude}DF . Find the given measurement. Explain your reasoning. (p. 319) E D F G 198 3 cm m FDG

Triangle DEG is isosceles with altitude}DF . Find the given measurement. Explain your reasoning. (p. 319) E D F G 198 3 cm FG

Sketch the indicated figure. Draw all of its lines of symmetry. (p. 619) Isosceles trapezoid

Sketch the indicated figure. Draw all of its lines of symmetry. (p. 619) Regular hexagon

Graph nABC. Then find its area. (p. 720) A(2, 2), B(9, 2), C(4, 16)

Graph nABC. Then find its area. (p. 720) A(28, 3), B(23, 3), C(21, 210)

VOCABULARY In Exercises 14, use the diagram shown. 8 5.5 6.8 G F B C A Identify the center of regular polygon ABCDE.

VOCABULARY In Exercises 14, use the diagram shown. 8 5.5 6.8 G F B C A Identify a central angle of the polygon.

VOCABULARY In Exercises 14, use the diagram shown. 8 5.5 6.8 G F B C A What is the radius of the polygon?

VOCABULARY In Exercises 14, use the diagram shown. 8 5.5 6.8 G F B C A What is the apothem?

WRITING Explain how to find the measure of a central angle of a regular polygon with n sides.

MEASURES OF CENTRAL ANGLES Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree, if necessary. 10 sides

MEASURES OF CENTRAL ANGLES Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree, if necessary. 18 sides

MEASURES OF CENTRAL ANGLES Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree, if necessary. 24 sides

MEASURES OF CENTRAL ANGLES Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree, if necessary. 7 sides

FINDING ANGLE MEASURES Find the given angle measure for the regular octagon shown. A B H C G K J D F E m GJH

FINDING ANGLE MEASURES Find the given angle measure for the regular octagon shown. A B H C G K J D F E m GJK

FINDING ANGLE MEASURES Find the given angle measure for the regular octagon shown. A B H C G K J D F E m KGJ

FINDING ANGLE MEASURES Find the given angle measure for the regular octagon shown. A B H C G K J D F E m EJH

FINDING AREA Find the area of the regular polygon. 12 2 3

FINDING AREA Find the area of the regular polygon. 6.84 1

FINDING AREA Find the area of the regular polygon. 2.5 2.77

ERROR ANALYSIS Describe and correct the error in finding the area of the regular hexagon. } 152 2 132 7.5 A 5 1 }2 a p ns A 5 1 }2 (13)(6)(7.5) 5 292.5 13 15

MULTIPLE CHOICE Which expression gives the apothem for a regular dodecagon with side length 8? A a 5} 4 tan 308 B a 5} 4 tan 158 C a 5} 8 tan 158 D a 5 8 p cos 158

PERIMETER AND AREA Find the perimeter and area of the regular polygon. 20

PERIMETER AND AREA Find the perimeter and area of the regular polygon. 4.1

PERIMETER AND AREA Find the perimeter and area of the regular polygon. 9

SHORT RESPONSE The perimeter of a regular nonagon is 18 inches. Is that enough information to find the area? If so, find the area and explain your steps. If not, explain why not.

CHOOSE A METHOD Identify any unknown length(s) you need to know to find the area of the regular polygon. Which methods in the table on page 764 can you use to find those lengths? Choose a method and find the area. 14

CHOOSE A METHOD Identify any unknown length(s) you need to know to find the area of the regular polygon. Which methods in the table on page 764 can you use to find those lengths? Choose a method and find the area. 10 10

CHOOSE A METHOD Identify any unknown length(s) you need to know to find the area of the regular polygon. Which methods in the table on page 764 can you use to find those lengths? Choose a method and find the area. 8.4 8

INSCRIBED SQUARE Find the area of the unshaded region in Exercise 23.

POLYGONS IN CIRCLES Find the area of the shaded region. 12

POLYGONS IN CIRCLES Find the area of the shaded region. 8

POLYGONS IN CIRCLES Find the area of the shaded region. 2 3 608

COORDINATE GEOMETRY Find the area of a regular pentagon inscribed in a circle whose equation is given by (x 2 4)2 1 (y 1 2)2 5 25.

REASONING Decide whether the statement is true or false. Explain. The area of a regular n-gon of fixed radius r increases as n increases.

REASONING Decide whether the statement is true or false. Explain. The apothem of a regular polygon is always less than the radius.

REASONING Decide whether the statement is true or false. Explain. The radius of a regular polygon is always less than the side length.

FORMULAS In Exercise 44 on page 726, the formula A 5 } 3s2 }4 for the area A of an equilateral triangle with side length s was developed. Show that the formulas for the area of a triangle and for the area of a regular polygon, A 5 1 }2 bh and A 5 1 }2 a p ns, also result in this formula when they are applied to an equilateral triangle with side length s.

CHALLENGE An equilateral triangle is shown inside a square inside a regular pentagon inside a regular hexagon. Write an expression for the exact area of the shaded regions in the figure. Then find the approximate area of the entire shaded region, rounded to the nearest whole unit. 8

BASALTIC COLUMNS Basaltic columns are geological formations that result from rapidly cooling lava. The Giants Causeway in Ireland, pictured here, contains many hexagonal columns. Suppose that one of the columns is in the shape of a regular hexagon with radius 8 inches. a. What is the apothem of the column? b. Find the perimeter and area of the column. Round the area to the nearest square inch.

WATCH A watch has a circular face on a background that is a regular octagon. Find the apothem and the area of the octagon. Then find the area of the silver border around the circular face. 0.2 cm 1 cm

COMPARING AREAS Predict which figure has the greatest area and which has the smallest area. Check by finding the area of each figure. a. 13 in. b. 15 in. 18 in. c. 9 in.

CRAFTS You want to make two wooden trivets, a large one and a small one. Both trivets will be shaped like regular pentagons. The perimeter of the small trivet is 15 inches, and the perimeter of the large trivet is 25 inches. Find the area of the small trivet. Then use the Areas of Similar Polygons Theorem to find the area of the large trivet. Round your answers to the nearest tenth.

CONSTRUCTION Use a ruler and compass. a. Draw }AB with a length of 1 inch. Open the compass to 1 inch and draw a circle with that radius. Using the same compass setting, mark off equal parts along the circle. Then connect the six points where the compass marks and circle intersect to draw a regular hexagon as shown. b. What is the area of the hexagon? of the shaded region? c. Explain how to construct an equilateral triangle. A B

HEXAGONS AND TRIANGLES Show that a regular hexagon can be divided into six equilateral triangles with the same side length.

ALTERNATIVE METHODS Find the area of a regular hexagon with side length 2 and apothem } 3 in at least four different ways.

APPLYING TRIANGLE PROPERTIES In Chapter 5, you learned properties of special segments in triangles. Use what you know about special segments in triangles to show that radius CP in equilateral nABC is twice the apothem DP. B A P D

EXTENDED RESPONSE Assume that each honeycomb cell is a regular hexagon. The distance is measured through the center of each cell. a. Find the average distance across a cell in centimeters. b. Find the area of a typical cell in square centimeters. Show your steps. c. What is the area of 100 cells in square centimeters? in square decimeters? (1 decimeter 5 10 centimeters.) d. Scientists are often interested in the number of cells per square decimeter. Explain how to rewrite your results in this form. 2.6 cm

CONSTANT PERIMETER Use a piece of string that is 60 centimeters long. a. Arrange the string to form an equilateral triangle and find the area. Next form a square and find the area. Then do the same for a regular pentagon, a regular hexagon, and a regular decagon. What is happening to the area? b. Predict and then find the areas of a regular 60-gon and a regular 120-gon. c. Graph the area A as a function of the number of sides n. The graph approaches a limiting value. What shape do you think will have the greatest area? What will that area be?

CHALLENGE Two regular polygons both have n sides. One of the polygons is inscribed in, and the other is circumscribed about, a circle of radius r. Find the area between the two polygons in terms of n and r

A jar contains 10 red marbles, 6 blue marbles, and 2 white marbles. Find the probability of the event described. (p. 893) You randomly choose one red marble from the jar, put it back in the jar, and then randomly choose a red marble.

A jar contains 10 red marbles, 6 blue marbles, and 2 white marbles. Find the probability of the event described. (p. 893) You randomly choose one blue marble from the jar, keep it, and then randomly choose one white marble.

Find the ratio of the width to the length of the rectangle. Then simplify the ratio. (p. 356) 9 ft 18 ft

Find the ratio of the width to the length of the rectangle. Then simplify the ratio. (p. 356) 42 cm 12 cm

Find the ratio of the width to the length of the rectangle. Then simplify the ratio. (p. 356) 36 in. 45 in

The vertices of quadrilateral ABCD are A(23, 3), B(1, 1), C(1, 23), and D(23, 21). Draw ABCD and determine whether it is a parallelogram. (p. 522)

VOCABULARY Copy and complete: If an event cannot occur, its probability is ? . If an event is certain to occur, its probability is ? .

WRITING Compare a geometric probability and a probability found by dividing the number of favorable outcomes by the total number of possible outcomes.

PROBABILITY ON A SEGMENT In Exercises 36, find the probability that a point K, selected randomly on }AE , is on the given segment. Express your answer as a fraction, decimal, and percent. 212 3 12 29 26 23 0 6 9 A DE AD

PROBABILITY ON A SEGMENT In Exercises 36, find the probability that a point K, selected randomly on }AE , is on the given segment. Express your answer as a fraction, decimal, and percent. 212 3 12 29 26 23 0 6 9 A DE BC

PROBABILITY ON A SEGMENT In Exercises 36, find the probability that a point K, selected randomly on }AE , is on the given segment. Express your answer as a fraction, decimal, and percent. 212 3 12 29 26 23 0 6 9 A DE }DE

PROBABILITY ON A SEGMENT In Exercises 36, find the probability that a point K, selected randomly on }AE , is on the given segment. Express your answer as a fraction, decimal, and percent. 212 3 12 29 26 23 0 6 9 A DE AE

WRITING Look at your answers to Exercises 3 and 5. Describe how the two probabilities are related.

FIND A GEOMETRIC PROBABILITY Find the probability that a randomly chosen point in the figure lies in the shaded region. 2 2

FIND A GEOMETRIC PROBABILITY Find the probability that a randomly chosen point in the figure lies in the shaded region. 14 6 12

FIND A GEOMETRIC PROBABILITY Find the probability that a randomly chosen point in the figure lies in the shaded region. 20 8 5

ERROR ANALYSIS Three sides of the rectangle are tangent to the semicircle. Describe and correct the error in finding the probability that a randomly chosen point in the figure lies in the shaded region. 10(7) 2 1 }2 p(5)2 }}10(7) 5} 70 2 12.5p 70 43.9%

ESTIMATING AREA Use the scale drawing. N W E S What is the approximate area of the north side of the island? the south side of the island? the whole island?

ESTIMATING AREA Use the scale drawing. N W E S Find the probability that a randomly chosen location on the island lies on the north side.

ESTIMATING AREA Use the scale drawing. N W E S Find the probability that a randomly chosen location on the island lies on the south side.

SIMILAR TRIANGLES In Exercise 9, how do you know that the shaded triangle is similar to the whole triangle? Explain how you can use the Area of Similar Polygons Theorem to find the desired probability.

ALGEBRA In Exercises 1619, find the probability that a point chosen at random on the segment satisfies the inequality. 2 9 345678 x 2 6 1

ALGEBRA In Exercises 1619, find the probability that a point chosen at random on the segment satisfies the inequality. 2 9 345679 1 2x 2 3 5

ALGEBRA In Exercises 1619, find the probability that a point chosen at random on the segment satisfies the inequality. 2 9 345680 x }2 7

ALGEBRA In Exercises 1619, find the probability that a point chosen at random on the segment satisfies the inequality. 2 9 345681 3x 27

FIND A GEOMETRIC PROBABILITY Find the probability that a randomly chosen point in the figure lies in the shaded region. Explain your steps. 5 7 3 3 13

FIND A GEOMETRIC PROBABILITY Find the probability that a randomly chosen point in the figure lies in the shaded region. Explain your steps. 14 12

FIND A GEOMETRIC PROBABILITY Find the probability that a randomly chosen point in the figure lies in the shaded region. Explain your steps. 8

MULTIPLE CHOICE A point X is chosen at random in region U, and U includes region A. What is the probability that X is not in A? A } Area of A Area of U B }} Area of A Area of U 2 Area of A C } 1 Area of A D Area of }}U 2 Area of A Area of U U A

ARCS AND SECTORS A sector of a circle intercepts an arc of 808. Find the probability that a randomly chosen point on the circle lies on the arc. Find the probability that a randomly chosen point in the circle lies in the sector. Explain why the probabilities do not depend on the radius.

INSCRIBED POLYGONS Find the probability that a randomly chosen point in the circle described lies in the inscribed polygon. Regular hexagon inscribed in circle with circumference C 188.5

INSCRIBED POLYGONS Find the probability that a randomly chosen point in the circle described lies in the inscribed polygon. Regular octagon inscribed in circle with radius r

INSCRIBED ANGLES Points A and B are the endpoints of a diameter of (D. Point C is chosen at random from the other points on the circle. What is the probability that nABC is a right triangle? What is the probability that m CAB 458? A B C D

COORDINATE GRAPHS Graph the system of inequalities 0 x 2, 0 y 3, and y x. If a point (x, y) is chosen at random in the solution region, what is the probability that x2 1 y 2 4?

CHALLENGE You carry out a series of steps to paint a walking stick. In the first step, you paint half the length of the stick. For each following step, you paint half of the remaining unpainted portion of the stick. After n steps, you choose a point at random on the stick. Find a value of n so that the probability of choosing a point on the painted portion of the stick after the nth step is greater than 99.95%.

DARTBOARD A dart is thrown and hits the target shown. If the dart is equally likely to hit any point on the target, what is the probability that it hits inside the inner square? that it hits outside the inner square but inside the circle? 18 in. 6

TRANSPORTATION A fair provides a shuttle bus from a parking lot to the fair entrance. Buses arrive at the parking lot every 10 minutes. They wait for 4 minutes while passengers get on and get off. Then the buses depart. 0 minutes wait time 2 4 6 8 10 a. What is the probability that there is a bus waiting when a passenger arrives at a random time? b. What is the probability that there is not a bus waiting when a passenger arrives at a random time?

FIRE ALARM Suppose that your school day is from 8:00 A.M. until 3:00 P.M. You eat lunch at 12:00 P.M. If there is a fire drill at a random time during the day, what is the probability that it begins before lunch?

PHONE CALL You are expecting a call from a friend anytime between 7:00 P.M. and 8:00 P.M. You are practicing the drums and cannot hear the phone from 6:55 P.M. to 7:10 P.M. What is the probability that you missed your friends call?

EXTENDED RESPONSE Scientists lost contact with the space probe Beagle 2 when it was landing on Mars in 2003. They have been unable to locate it since. Early in the search, some scientists thought that it was possible, though unlikely, that Beagle had landed in a circular crater inside the planned landing region. The diameter of the crater is 1 km. a. In the scale drawing, each square has side length 2 kilometers. Estimate the area of the planned landing region. Explain your steps. b. Estimate the probability of Beagle 2 landing in the crater if it was equally likely to land anywhere in the planned landing region.

SHORT RESPONSE If the central angle of a sector of a circle stays the same and the radius of the circle doubles, what can you conclude about the probability of a randomly selected point being in the sector? Explain. Include an example with your explanation.

PROBABILITY AND LENGTH A 6 inch long rope is cut into two pieces at a random point. Find the probability both pieces are at least 1 inch long.

COMPOUND EVENTS You throw two darts at the dartboard in Exercise 30 on page 776. Each dart hits the dartboard. The throws are independent of each other. Find the probability of the compound event described. a. Both darts hit the yellow square. b. The first dart hits the yellow square and the second hits outside the circle. c. Both darts hit inside the circle but outside the yellow square.

CHALLENGE A researcher used a 1 hour tape to record birdcalls. Eight minutes after the recorder was turned on, a 5 minute birdcall began. Later, the researcher accidentally erased 10 continuous minutes of the tape. What is the probability that part of the birdcall was erased? What is the probability that all of the birdcall was erased?

Draw a concave hexagon and a concave pentagon. (p. 42)

Think of each segment shown as part of a line. B A F H G D Name the intersection of plane DCH and plane ADE. (p. 96)

Think of each segment shown as part of a line. B A F H G D Name a plane that appears to be parallel to plane ADH. (p. 147)

Find the area of the polygon. (p. 720) 0.3 m 0.6 m

Find the area of the polygon. (p. 730) 9 ft 12 ft 7 ft

Find the area of the polygon. (p. 762) 4 in. 4.6 in

Copy and complete: A sector of a circle is the region bounded by ? .

WRITING Explain the relationship between the height of a parallelogram and the bases of a parallelogram.

The diagram shows a square inscribed in a circle. Name an example of the given segment. Z X Y An apothem of the square

The diagram shows a square inscribed in a circle. Name an example of the given segment. Z X Y A radius of the square

Find the area of the polygon. 6 10 7

Find the area of the polygon. 68 3

Find the area of the polygon. 16 40

The area of a triangle is 147 square inches and its height is 1.5 times its base. Find the base and the height of the triangle.

Graph the polygon with the given vertices and find its area. L(2, 2), M(6, 2), N(8, 4), P(4, 4)

Graph the polygon with the given vertices and find its area. Q(23, 0), R(22, 3), S(21, 0), T(22, 22)

Graph the polygon with the given vertices and find its area. D(21, 4), E(5, 4), F(3, 22), G(1, 22)

The polygons are similar. Find the ratio (red to blue) of the perimeters and of the areas. Then find the unknown area. nABC , nDEF F B E A 3 ft D 4 ft A

The polygons are similar. Find the ratio (red to blue) of the perimeters and of the areas. Then find the unknown area. WXYZ , ABCD 10 cm A D B C 13 cm A

The ratio of the areas of two similar figures is 144 : 49. Write the ratio of the lengths of corresponding sides.

Find the indicated measure. Diameter of (F F C 5 94.24 ft

Find the indicated measure. Circumference of (F 358 F H G 5.50 cm

Find the indicated measure. Length ofCGH 1158 F H

Find the area of the blue shaded region. 2408 V W

Find the area of the blue shaded region. 6 in. 4 in.

Find the area of the blue shaded region. 27.93 ft2 S 508 T R

PLATTER A platter is in the shape of a regular octagon. Find the perimeter and area of the platter if its apothem is 6 inches.

PUZZLE A jigsaw puzzle is in the shape of a regular pentagon. Find its area if its radius is 17 centimeters and its side length is 20 centimeters.

A point K is selected randomly on }AC at the right. What is the probability that K is on }AB ? 22 21 A B 01234 5

Find the probability that a randomly chosen point in the figure lies in the shaded region. 258 12

Find the probability that a randomly chosen point in the figure lies in the shaded region. 15 6 18

Find the probability that a randomly chosen point in the figure lies in the shaded region. 4 4