PROOF Use the given information to draw a sketch. Then write a proof. GIVEN c nSTU

VOCABULARY Copy the proportion m }n 5 p }q . Identify the means of the proportion and the extremes of the proportion.

WRITING Write three ratios that are equivalent to the ratio 3 : 4. Explain how you found the ratios.

SIMPLIFYING RATIOS Simplify the ratio. $20 : $5

SIMPLIFYING RATIOS Simplify the ratio. 15 cm2 12 cm2

SIMPLIFYING RATIOS Simplify the ratio. 6 L : 10 mL

SIMPLIFYING RATIOS Simplify the ratio. 1 mi 20 ft

SIMPLIFYING RATIOS Simplify the ratio. 7 ft 12 in

SIMPLIFYING RATIOS Simplify the ratio. 80 cm 2 m

SIMPLIFYING RATIOS Simplify the ratio. 3 lb 10 oz

SIMPLIFYING RATIOS Simplify the ratio. 2 gallons 18 quarts

WRITING RATIOS Find the ratio of the width to the length of the rectangle. Then simplify the ratio. 15 in. 5 in

WRITING RATIOS Find the ratio of the width to the length of the rectangle. Then simplify the ratio. 16 cm 18 cm

WRITING RATIOS Find the ratio of the width to the length of the rectangle. Then simplify the ratio. 10 m 320 cm

FINDING RATIOS Use the number line to find the ratio of the distances. 0 7 1 45 2 8 16 3 6 9 12 13 10 11 14 15 A AD CF

FINDING RATIOS Use the number line to find the ratio of the distances. 0 7 1 45 2 8 16 3 6 9 12 13 10 11 14 15 A BD AB

FINDING RATIOS Use the number line to find the ratio of the distances. 0 7 1 45 2 8 16 3 6 9 12 13 10 11 14 15 A CE EF

FINDING RATIOS Use the number line to find the ratio of the distances. 0 7 1 45 2 8 16 3 6 9 12 13 10 11 14 15 A BE CE

PERIMETER The perimeter of a rectangle is 154 feet. The ratio of the length to the width is 10 : 1. Find the length and the width.

SEGMENT LENGTHS In the diagram, AB:BC is 2 : 7 and AC 5 36. Find AB and BC. 36 A B C

USING EXTENDED RATIOS The measures of the angles of a triangle are in the extended ratio given. Find the measures of the angles of the triangle. 3 : 5 : 10

USING EXTENDED RATIOS The measures of the angles of a triangle are in the extended ratio given. Find the measures of the angles of the triangle. 2:7:9

USING EXTENDED RATIOS The measures of the angles of a triangle are in the extended ratio given. Find the measures of the angles of the triangle. 11 : 12 : 13

ALGEBRA Solve the proportion. 6 }x 5 3 }2

ALGEBRA Solve the proportion. y }20 5 3 }10

ALGEBRA Solve the proportion. 2 }7 5 12 }z

ALGEBRA Solve the proportion. j 1 1 }5 5 4 }10

ALGEBRA Solve the proportion. 1 c 1 5 5 3 }24

ALGEBRA Solve the proportion. 4 a 2 3 5 2 }5

ALGEBRA Solve the proportion. 1 } 1 3b 4 5 5 }2

ALGEBRA Solve the proportion. 3 2p 1 5 5 1 }9p

GEOMETRIC MEAN Find the geometric mean of the two numbers. 2 and 18

GEOMETRIC MEAN Find the geometric mean of the two numbers. 4 and 25

GEOMETRIC MEAN Find the geometric mean of the two numbers. 32 and 8

GEOMETRIC MEAN Find the geometric mean of the two numbers. 4 and 16

GEOMETRIC MEAN Find the geometric mean of the two numbers. 2 and 25

GEOMETRIC MEAN Find the geometric mean of the two numbers. 6 and 20

ERROR ANALYSIS A student incorrectly simplified the ratio. Describe and correct the students error. 8 in. 3 ft 5 } 8 in. 3 ft p } 12 in. 1 ft 5} 96 in. 3 ft 5} 32 in. 1 ft

WRITING RATIOS Let x 5 10, y 5 3, and z 5 8. Write the ratio in simplest form. x : z

WRITING RATIOS Let x 5 10, y 5 3, and z 5 8. Write the ratio in simplest form. 8y }x

WRITING RATIOS Let x 5 10, y 5 3, and z 5 8. Write the ratio in simplest form. 4 2x 1 2z

WRITING RATIOS Let x 5 10, y 5 3, and z 5 8. Write the ratio in simplest form. 2 } x 2 z 3y

ALGEBRA Solve the proportion. 2 } x 1 5 3 5 x } 2 5 4

ALGEBRA Solve the proportion. 2 } 2 s 3 5 2 } s 1 1 5

ALGEBRA Solve the proportion. 15 }m 5 m }5

ALGEBRA Solve the proportion. 7 q 1 1 5 q } 2 1 5

ANGLE MEASURES The ratio of the measures of two supplementary angles is 5 : 3. Find the measures of the angles.

SHORT RESPONSE The ratio of the measure of an exterior angle of a triangle to the measure of the adjacent interior angle is 1 : 4. Is the triangle acute or obtuse? Explain how you found your answer.

SHORT RESPONSE Without knowing its side lengths, can you determine the ratio of the perimeter of a square to the length of one of its sides? Explain.

ALGEBRA In Exercises 4951, the ratio of two side lengths for the triangle is given. Solve for the variable. AB:BC is 3 : 8. 24 C B A

ALGEBRA In Exercises 4951, the ratio of two side lengths for the triangle is given. Solve for the variable. AB:BC is 3 : 4. A 10q 1 15

ALGEBRA In Exercises 4951, the ratio of two side lengths for the triangle is given. Solve for the variable. AB:BC is 5 : 9. A 5r 1 2

MULTIPLE CHOICE What is a value of x that makes x }3 5} 4x x 1 3 true? A 3 B 4 C 9 D 12

AREA The area of a rectangle is 4320 square inches. The ratio of the width to the length is 5 : 6. Find the length and the width.

COORDINATE GEOMETRY The points (23, 2), (1, 1), and (x, 0) are collinear. Use slopes to write a proportion to find the value of x.

ALGEBRA Use the proportions } a 1 b 2a 2 b 5 5 }4 and} b a 1 9 5 5 }9 to find a and b.

CHALLENGE Find the ratio of x to y given that 5 }y 1 7 }x 5 24 and 12 }y 1 2 }x 5 24.

TILING The perimeter of a room is 66 feet. The ratio of its length to its width is 6 : 5. You want to tile the floor with 12 inch square tiles. Find the length and width of the room, and the area of the floor. How many tiles will you need? The tiles cost $1.98 each. What is the total cost to tile the floor?

GEARS The gear ratio of two gears is the ratio of the number of teeth of the larger gear to the number of teeth of the smaller gear. In a set of three gears, the ratio of Gear A to Gear B is equal to the ratio of Gear B to Gear C. Gear A has 36 teeth and Gear C has 16 teeth. How many teeth does Gear B have?

TRAIL MIX You need to make 36 one-half cup bags of trail mix for a class trip. The recipe calls for peanuts, chocolate chips, and raisins in the extended ratio 5 : 1 : 4. How many cups of each item do you need?

PAPER SIZES International standard paper sizes are commonly used all over the world. The various sizes all have the same width-tolength ratios. Two sizes of paper are shown, called A3 and A2. The distance labeled x is the geometric mean of 297 mm and 594 mm. Find the value of x. 297 mm 594 mm A

BATTING AVERAGE The batting average of a baseball player is the ratio of the number of hits to the number of official at-bats. In 2004, Johnny Damon of the Boston Red Sox had 621 official at-bats and a batting average of .304. Use the proportion to find the number of hits made by Johnny Damon. }} Number of hits Number of at-bats 5}} Batting average 1.000

MULTI-STEP PROBLEM The population of Red-tailed hawks is increasing in many areas of the United States. One long-term survey of bird populations suggests that the Red-tailed hawk population is increasing nationally by 2.7% each year. a. Write 2.7% as a ratio of hawks in year n to hawks in year (n 2 1). b. In 2004, observers in Corpus Christi, TX, spotted 180 migrating Redtailed hawks. Assuming this population follows the national trend, about how many Red-tailed hawks can they expect to see in 2005? c. Observers in Lipan Point, AZ, spotted 951 migrating Red-tailed hawks in 2004. Assuming this population follows the national trend, about how many Red-tailed hawks can they expect to see in 2006?

SHORT RESPONSE Some common computer screen resolutions are 1024 : 768, 800 : 600, and 640 : 480. Explain why these ratios are equivalent.

BIOLOGY The larvae of the Mother-of-Pearl moth is the fastest moving caterpillar. It can run at a speed of 15 inches per second. When threatened, it can curl itself up and roll away 40 times faster than it can run. How fast can it run in miles per hour? How fast can it roll?

CURRENCY EXCHANGE Emily took 500 U.S. dollars to the bank to exchange for Canadian dollars. The exchange rate on that day was 1.2 Canadian dollars per U.S. dollar. How many Canadian dollars did she get in exchange for the 500 U.S. dollars?

MULTIPLE REPRESENTATIONS Let x and y be two positive numbers whose geometric mean is 6. a. Making a Table Make a table of ordered pairs (x, y) such that } xy 5 6. b. Drawing a Graph Use the ordered pairs to make a scatter plot. Connect the points with a smooth curve. c. Analyzing Data Is the data linear? Why or why not?

ALGEBRA Use algebra to verify Property 1, the Cross Products Property.

ALGEBRA Show that the geometric mean of two numbers is equal to the arithmetic mean (or average) of the two numbers only when the numbers are equal. (Hint: Solve } xy 5 x } 1 y 2 with x, y 0.)

CHALLENGE In Exercises 6971, use the given information to find the value(s) of x. Assume that the given quantities are nonnegative. The geometric mean of the quantities (} x) and (3} x) is (x 2 6).

CHALLENGE In Exercises 6971, use the given information to find the value(s) of x. Assume that the given quantities are nonnegative. The geometric mean of the quantities (x 1 1) and (2x 1 3) is (x 1 3).

CHALLENGE In Exercises 6971, use the given information to find the value(s) of x. Assume that the given quantities are nonnegative. The geometric mean of the quantities (2x 1 1) and (6x 1 1) is (4x 2 1).

Find the reciprocal. (p. 869) 26

Find the reciprocal. (p. 869) 1 }13

Find the reciprocal. (p. 869) 236 3

Find the reciprocal. (p. 869) 20.2

Solve the quadratic equation. (p. 882) 5x2 5 35

Solve the quadratic equation. (p. 882) x2 2 20 5 29

Solve the quadratic equation. (p. 882) (x 2 3)(x 1 3) 5 27

Write the equation of the line with the given description. (p. 180) Parallel to y 5 3x 2 7, passing through (1, 2)

Write the equation of the line with the given description. (p. 180) Perpendicular to y 5 1 }4 x 1 5, passing through (0, 24)

VOCABULARY Copy and complete: A ? is a drawing that has the same shape as the object it represents.

WRITING Suppose the scale of a model of the Eiffel Tower is 1 inch : 20 feet. Explain how to determine how many times taller the actual tower is than the model.

REASONING Copy and complete the statement. If 8 }x 5 3 }y , then 8 }3 5 ? }?

REASONING Copy and complete the statement. If x }9 5 y }20 , then x }y 5 ? }?

REASONING Copy and complete the statement. If x }6 5 y }15 , then x } 1 6 6 5 ? }?

REASONING Copy and complete the statement. If 14 }3 5 x }y , then 17 }3 5 ? }?

REASONING Decide whether the statement is true or false. If 8 }m 5 n }9 , then 8 } 1 m m 5 n } 1 9 9

REASONING Decide whether the statement is true or false. If 5 }7 5 a }b , then 7 }5 5 a }b

REASONING Decide whether the statement is true or false. If d }2 5 g } 1 10 11 , then } d g 1 10 5 2 }11 .

REASONING Decide whether the statement is true or false. f 4 } 1 x 4 5 3 } 1 y y , then x }4 5 3 }y

PROPERTIES OF PROPORTIONS Use the diagram and the given information to find the unknown length. Given } CB BA 5 } DE EF , find BA. 7 6 4 A F B E C

PROPERTIES OF PROPORTIONS Use the diagram and the given information to find the unknown length. Given } XW XV 5 } YW ZV , find ZV V Y W X

SCALE DIAGRAMS In Exercises 13 and 14, use the diagram of the field hockey field in which 1 inch 5 50 yards. Use a ruler to approximate the dimension. Find the actual length of the field.

SCALE DIAGRAMS In Exercises 13 and 14, use the diagram of the field hockey field in which 1 inch 5 50 yards. Use a ruler to approximate the dimension. Find the actual width of the field.

ERROR ANALYSIS Describe and correct the error made in the reasoning. If a }3 5 c }4 , then a } 1 3 3 5 c } 1 3 4 .

PROPERTIES OF PROPORTIONS Use the diagram and the given information to find the unknown length. Given } CA CB 5 } AE BD , find BD. 7 14 18 A E B D C

PROPERTIES OF PROPORTIONS Use the diagram and the given information to find the unknown length. Given SQ }SR 5 } TV TU , find RQ. V 3 7 7 S R

MULTIPLE CHOICE If x, y, z, and q are four different numbers, and the proportion x }y 5 z }q is true, which of the following is false? A y }x 5 q }z B x }z 5 y }q C y }x 5 z }q D x } 1 y y 5 z } 1 q q

CHALLENGE Two number patterns are proportional if there is a nonzero number k such that (a1, b1, c1, . . .) 5 k(a2, b2, c2, . . .) 5 ka2, kb2, kc2, . . .. Given the relationship (8, 16, 20) 5 k(2, 4, 5), find k.

CHALLENGE Two number patterns are proportional if there is a nonzero number k such that (a1, b1, c1, . . .) 5 k(a2, b2, c2, . . .) 5 ka2, kb2, kc2, . . .. Given that a1 5 ka2, b1 5 kb2, and c1 5 kc2, show that a1 }a2 5 b1 }b2 5 c1 }c2

CHALLENGE Two number patterns are proportional if there is a nonzero number k such that (a1, b1, c1, . . .) 5 k(a2, b2, c2, . . .) 5 ka2, kb2, kc2, . . .. Given that a1 5 ka2, b1 5 kb2, and c1 5 kc2, show that a1 1 b1 1 c }1 a2 1 b2 1 c2 5 k.

ARCHITECTURE A basket manufacturer has headquarters in an office building that has the same shape as a basket they sell. a. The bottom of the basket is a rectangle with length 15 inches and width 10 inches. The base of the building is a rectangle with length 192 feet. What is the width of the base of the building? b. About how many times as long as the bottom of the basket is the base of the building? Longaberger Company Home Office Newark, Ohio

MAP SCALE A street on a map is 3 inches long. The actual street is 1 mile long. Find the scale of the m

MULTIPLE CHOICE A model train engine is 12 centimeters long. The actual engine is 18 meters long. What is the scale of the model? A 3 cm : 2 m B 1 cm : 1.5 m C 1 cm : 3 m D 200 cm : 3 m

MAP READING The map of a hiking trail has a scale of 1 inch : 3.2 miles. Use a ruler to approximate the actual distance between the two shelters. Meadow View Whispering Pines Blueberry Hill Meadow View and Whispering Pines

MAP READING The map of a hiking trail has a scale of 1 inch : 3.2 miles. Use a ruler to approximate the actual distance between the two shelters. Meadow View Whispering Pines Blueberry Hill Whispering Pines and Blueberry Hill

POLLEN The photograph shows a particle of goldenrod pollen that has been magnified under a microscope. The scale of the photograph is 900 : 1. Use a ruler to estimate the width in millimeters of the particle.

RAMP DESIGN Assume that the wheelchair ramps described each have a slope of 1 }12 , which is the maximum slope recommended for a wheelchair ramp. A wheelchair ramp has a 21 foot run. What is its rise?

RAMP DESIGN Assume that the wheelchair ramps described each have a slope of 1 }12 , which is the maximum slope recommended for a wheelchair ramp. A wheelchair ramp rises 4 feet. What is its run?

STATISTICS Researchers asked 4887 people to pick a number between 1 and 10. The results are shown in the table below. Answer 12345 Percent 4.2% 5.1% 11.4% 10.5% 10.7% Answer 6 7 8 9 10 Percent 10.0% 27.2% 8.8% 6.0% 6.1%a. Estimate the number of people who picked the number 3. b. You ask a participant what number she picked. Is the participant more likely to answer 6 or 7? Explain. c. Conduct this experiment with your classmates. Make a table in which you compare the new percentages with the ones given in the original survey. Why might they be different?

ALGEBRA Use algebra to verify the property of proportions. Property 2

ALGEBRA Use algebra to verify the property of proportions. Property 3

ALGEBRA Use algebra to verify the property of proportions. Property 4

REASONING Use algebra to explain why the property of proportions is true. If a } 2 b a 1 b 5 c } 2 d c 1 d , then a }b 5 c }d.

REASONING Use algebra to explain why the property of proportions is true. If } a 1 c b 1 d 5 } a 2 c b 2 d , then a }b 5 c }d.

REASONING Use algebra to explain why the property of proportions is true. If a }b 5 c }d 5 e }f , then a } 1 c 1 e b 1 d 1 f 5 a }b . (Hint: Let a }b 5 r.)

CHALLENGE When fruit is dehydrated, water is removed from the fruit. The water content in fresh apricots is about 86%. In dehydrated apricots, the water content is about 75%. Suppose 5 kilograms of raw apricots are dehydrated. How many kilograms of water are removed from the fruit? What is the approximate weight of the dehydrated apricots?

Over the weekend, Claudia drove a total of 405 miles, driving twice as far on Saturday as on Sunday. How far did Claudia travel each day? (p. 65)

Identify all pairs of congruent corresponding parts. Then write another congruence statement for the figures. (p. 225) nXYZ > nLMN X Y M N L

Identify all pairs of congruent corresponding parts. Then write another congruence statement for the figures. (p. 225) DEFG > QRST D F R P E S T

VOCABULARY Copy and complete: Two polygons are similar if corresponding angles are ? and corresponding side lengths are ? .

WRITING If two polygons are congruent, must they be similar? If two polygons are similar, must they be congruent? Explain.

USING SIMILARITY List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality. nABC , nLMN N M L C B A

USING SIMILARITY List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality. DEFG , PQRS P P S R G D

USING SIMILARITY List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality. HJKL , WXYZ H J Y K W

MULTIPLE CHOICE Triangles ABC and DEF are similar. Which statement is not correct? A } BC EF 5 } BC EF B } AB DE 5 } CA FD C } CA FD 5 } BC EF D } AB EF 5 } BC DE

DETERMINING SIMILARITY Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor. R S U 64 64 48 48 X Z W Y 24 24 32 3

DETERMINING SIMILARITY Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor. C D T 12 8 5 4 10 9

USING SIMILAR POLYGONS In the diagram, JKLM , EFGH. K J M x L 30 3 20 658 E F H Find the scale factor of JKLM to EFGH.

USING SIMILAR POLYGONS In the diagram, JKLM , EFGH. K J M x L 30 3 20 658 E F H Find the values of x, y, and z.

USING SIMILAR POLYGONS In the diagram, JKLM , EFGH. K J M x L 30 3 20 658 E F H Find the perimeter of each polygon.

PERIMETER Two similar FOR SALE signs have a scale factor of 5 : 3. The large signs perimeter is 60 inches. Find the small signs perimeter.

ERROR ANALYSIS The triangles are similar. Describe and correct the error in finding the perimeter of Triangle B. 10 6 12 B 5 Perimeter of B 5 56

REASONING Are the polygons always, sometimes, or never similar? Two isosceles triangles

REASONING Are the polygons always, sometimes, or never similar? Two equilateral triangles

REASONING Are the polygons always, sometimes, or never similar? A right triangle and an isosceles triangle

REASONING Are the polygons always, sometimes, or never similar? A scalene triangle and an isosceles triangle

SHORT RESPONSE The scale factor of Figure A to Figure B is 1 : x. What is the scale factor of Figure B to Figure A? Explain your reasoning.

SIMILAR TRIANGLES Identify the type of special segment shown in blue, and find the value of the variable. 27 x 16 18

SIMILAR TRIANGLES Identify the type of special segment shown in blue, and find the value of the variable. 18 16 y y

USING SCALE FACTOR Triangles NPQ and RST are similar. The side lengths of nNPQ are 6 inches, 8 inches, and 10 inches, and the length of an altitude is 4.8 inches. The shortest side of nRST is 8 inches long. Find the lengths of the other two sides of nRST.

USING SCALE FACTOR Triangles NPQ and RST are similar. The side lengths of nNPQ are 6 inches, 8 inches, and 10 inches, and the length of an altitude is 4.8 inches. The shortest side of nRST is 8 inches long. Find the length of the corresponding altitude in nRST.

USING SIMILAR TRIANGLES In the diagram, nABC , nDEF. D F E B 19 4 5 10 2 5 Find the scale factor of nABC to nDEF.

USING SIMILAR TRIANGLES In the diagram, nABC , nDEF. D F E B 19 4 5 10 2 5 Find the unknown side lengths in both triangles.

USING SIMILAR TRIANGLES In the diagram, nABC , nDEF. D F E B 19 4 5 10 2 5 Find the length of the altitude shown in nABC.

USING SIMILAR TRIANGLES In the diagram, nABC , nDEF. D F E B 19 4 5 10 2 5 Find and compare the areas of both triangles.

SHORT RESPONSE Suppose you are told that nPQR , nXYZ and that the extended ratio of the angle measures in nPQR is x : x 1 30 : 3x. Do you need to know anything about nXYZ to be able to write its extended ratio of angle measures? Explain your reasoning.

MULTIPLE CHOICE The lengths of the legs of right triangle ABC are 3 feet and 4 feet. The shortest side of nUVW is 4.5 feet and nUVW , nABC. How long is the hypotenuse of nUVW? A 1.5 ft B 5 ft C 6 ft D 7.5 ft

CHALLENGE Copy the figure at the right and divide it into two similar figures.

REASONING Is similarity reflexive? symmetric? transitive? Give examples to support your answers.

TENNIS In table tennis, the table is a rectangle 9 feet long and 5 feet wide. A tennis court is a rectangle 78 feet long and 36 feet wide. Are the two surfaces similar? Explain. If so, find the scale factor of the tennis court to the table.

DIGITAL PROJECTOR You are preparing a computer presentation to be digitally projected onto the wall of your classroom. Your computer screen is 13.25 inches wide and 10.6 inches high. The projected image on the wall is 53 inches wide and 42.4 inches high. Are the two shapes similar? If so, find the scale factor of the computer screen to the projected image.

MULTIPLE REPRESENTATIONS Use the similar figures shown. The scale factor of Figure 1 to Figure 2 is 7 : 10. a. Making a Table Copy and complete the table. AB BC CD DE EA Figure 1 3.5 ? ? ? ? Figure 2 5.0 4.0 6.0 8.0 3.0 b. Drawing a Graph Graph the data in the table. Let x represent the length of a side in Figure 1 and let y represent the length of the corresponding side in Figure 2. Is the relationship linear? c. Writing an Equation Write an equation that relates x and y. What is its slope? How is the slope related to the scale factor? D B C E A 6 8 4 3 5 D B

MULTI-STEP PROBLEM During a total eclipse of the sun, the moon is directly in line with the sun and blocks the suns rays. The distance ED between Earth and the moon is 240,000 miles, the distance DA between Earth and the sun is 93,000,000 miles, and the radius AB of the sun is 432,500 miles. a. Copy the diagram and label the known distances. b. In the diagram, nBDA , nCDE. Use this fact to explain a total eclipse of the sun. c. Estimate the radius CE of the moon. -OON %ARTH ! "

SHORT RESPONSE A rectangular image is enlarged on each side by the same amount. The angles remain unchanged. Can the larger image be similar to the original? Explain your reasoning, and give an example to support your answer. w w 1 a l 1 a

SHORT RESPONSE How are the areas of similar rectangles related to the scale factor? Use examples to justify your reasoning.

EXTENDED RESPONSE The equations of two lines in the coordinate plane are y 5 4 }3 x 1 4 and y 5 4 }3 x 2 8. a. Explain why the two lines are parallel. b. Show that BOA > DOC, OBA > ODC, and BAO > DCO. c. Find the coordinates of points A, B, C, and D. Find the lengths of the sides of nAOB and nCOD. d. Show that nAOB , nCOD. y D A B C

PROVING THEOREM 6.1 Prove the Perimeters of Similar Polygons Theorem for similar rectangles. Include a diagram in your proof.

CHALLENGE In the diagram, PQRS is a square, and PLMS , LMRQ. Find the exact value of x. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. Show that the similar rectangles in the diagram are golden rectangles. R M P L S P

Given A(1, 1), B(3, 2), C(2, 4), and D11, 7 }22 , determine whether the following lines are parallel, perpendicular, or neither. (p. 171) AB and ] BC

Given A(1, 1), B(3, 2), C(2, 4), and D11, 7 }22 , determine whether the following lines are parallel, perpendicular, or neither. (p. 171) CD and ] AD

Given A(1, 1), B(3, 2), C(2, 4), and D11, 7 }22 , determine whether the following lines are parallel, perpendicular, or neither. (p. 171) ] AB and ] CD

Find the measure of the exterior angle shown. (p. 217) 3. 1 (3x 2 50)8 x

Find the measure of the exterior angle shown. (p. 218) (2x 1 20)8 408 x8

Find the measure of the exterior angle shown. (p. 219) (3x 1 8)8 [6(x 2 2)]8 25

Copy and complete the statement with <, >, or 5. (p. 335) RS ? TU 1158 1108 U V R

Copy and complete the statement with <, >, or 5. (p. 335) FG ? HD H E F D

Copy and complete the statement with <, >, or 5. (p. 335) WX ? YX W Y Z X 488 4

VOCABULARY Copy and complete: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are ? .

WRITING Can you assume that corresponding sides and corresponding angles of any two similar triangles are congruent? Explain.

REASONING Use the diagram to complete the statement. A C E 25 15 F D x y 18 1 nABC , ?

REASONING Use the diagram to complete the statement. A C E 25 15 F D x y 18 1 BA ? 5 } AC ? 5 } CB ?

REASONING Use the diagram to complete the statement. A C E 25 15 F D x y 18 1 25 }? 5 ? }12

REASONING Use the diagram to complete the statement. A C E 25 15 F D x y 18 1 ? }25 5 18 }?

REASONING Use the diagram to complete the statement. A C E 25 15 F D x y 18 1 y 5 ?

REASONING Use the diagram to complete the statement. A C E 25 15 F D x y 18 1 x 5 ?

AA SIMILARITY POSTULATE In Exercises 914, determine whether the triangles are similar. If they are, write a similarity statement. F G H L K J 428

AA SIMILARITY POSTULATE In Exercises 914, determine whether the triangles are similar. If they are, write a similarity statement. M Y Z 458

AA SIMILARITY POSTULATE In Exercises 914, determine whether the triangles are similar. If they are, write a similarity statement. R P V T 858 658 358 358

AA SIMILARITY POSTULATE In Exercises 914, determine whether the triangles are similar. If they are, write a similarity statement. D A

AA SIMILARITY POSTULATE In Exercises 914, determine whether the triangles are similar. If they are, write a similarity statement. W U Y Z X 508 458 858

AA SIMILARITY POSTULATE In Exercises 914, determine whether the triangles are similar. If they are, write a similarity statement. P N M L

ERROR ANALYSIS Explain why the students similarity statement is incorrect. ABCD , EFGH by AA Similarity Postulate C E G D H A B F

MULTIPLE CHOICE What is the value of p? A 5 B 20 C 28.8 D Cannot be determined 10 24 p

ERROR ANALYSIS A student uses the proportion 4 }6 5 5 }x to find the value of x in the figure. Explain why this proportion is incorrect and write a correct proportion. 5 4 6 x

OPEN-ENDED MATH In Exercises 18 and 19, make a sketch that can be used to show that the statement is false. If two pairs of sides of two triangles are congruent, then the triangles are similar.

OPEN-ENDED MATH In Exercises 18 and 19, make a sketch that can be used to show that the statement is false. If the ratios of two pairs of sides of two triangles are proportional, then the triangles are similar.

MULTIPLE CHOICE In the figure at the right, find the length of }BD . A 35 }3 B 37 }5 C 20 }3 D 12 }5 3 4 5 x A

ALGEBRA Find coordinates for point E so that nABC S nADE. D B A CE A(0, 0), B(0, 4), C(8, 0), D(0, 5), E(x, y)

ALGEBRA Find coordinates for point E so that nABC S nADE. D B A CE A(0, 0), B(0, 3), C(4, 0), D(0, 7), E(x, y)

ALGEBRA Find coordinates for point E so that nABC S nADE. D B A CE A(0, 0), B(0, 1), C(6, 0), D(0, 4), E(x, y)

ALGEBRA Find coordinates for point E so that nABC S nADE. D B A CE A(0, 0), B(0, 6), C(3, 0), D(0, 9), E(x, y)

MULTI-STEP PROBLEM In the diagram, ] AB i ] DC , AE 5 6, AB 5 8, CE 5 15, and DE 5 10. a. Copy the diagram and mark all given information. b. List two pairs of congruent angles in the diagram. c. Name a pair of similar triangles and write a similarity statement. d. Find BE and DC. E D C A

REASONING In Exercises 2629, is it possible for nJKL and nXYZ to be similar? Explain why or why not. m J 5 718, m K 5 528, m X 5 718, and m Z 5 578

REASONING In Exercises 2629, is it possible for nJKL and nXYZ to be similar? Explain why or why not. nJKL is a right triangle and m X 1 m Y 5 1508.

REASONING In Exercises 2629, is it possible for nJKL and nXYZ to be similar? Explain why or why not. m J 5 878 and m Y 5 948

REASONING In Exercises 2629, is it possible for nJKL and nXYZ to be similar? Explain why or why not. m J 1 m K 5 858 and m Y 1 m Z 5 808

HALLENGE If PT 5 x, PQ 5 3x, and SR 5 8 }3 x, find PS in terms of x. Explain your reasoning. R S T P P

AIR HOCKEY An air hockey player returns the puck to his opponent by bouncing the puck off the wall of the table as shown. From physics, the angles that the path of the puck makes with the wall are congruent. What is the distance d between the puck and the wall when the opponent returns it? 20 in. 26 in. 66 in. puck wall

LAKES You can measure the width of the lake using a surveying technique, as shown in the diagram. a. What postulate or theorem can you use to show that the triangles are similar? b. Find the width of the lake, WX. c. If XY 5 10 meters, find VX. V X 104 m 6 m 8 m

SHORT RESPONSE Explain why all equilateral triangles are similar. Include sketches in your answer.

AERIAL PHOTOGRAPHY Low-level aerial photos can be taken using a remote-controlled camera suspended from a blimp. You want to take an aerial photo that covers a ground distance g of 50 meters. Use the proportion f }h 5 n }g to estimate the altitude h that the blimp should fly at to take the photo. In the proportion, use f 5 8 centimeters and n 5 3 centimeters. These two variables are determined by the type of camera used.

PROOF Use the given information to draw a sketch. Then write a proof. GIVEN c nSTU , nPQR Point V lies on }TU so that }SV bisects TSU. Point N lies on }QR so that }PN bisects QPR. PROVE c } SV PN 5 } ST PQ

PROOF Prove that if an acute angle in one right triangle is congruent to an acute angle in another right triangle, then the triangles are similar.

TECHNOLOGY Use a graphing calculator or computer. a. Draw nABC. Draw }DE through two sides of the triangle, parallel to the third side. b. Measure ADE and ACB. Measure AED and ABC. What do you notice? c. What does a postulate in this lesson tell you about nADE and nACB? d. Measure all the sides. Show that corresponding side lengths are proportional. e. Move vertex A to form new triangles. How do your measurements in parts (b) and (d) change? Are the new triangles still similar? Explain. B A D

EXTENDED RESPONSE Explain how you could use similar triangles to show that any two points on a line can be used to calculate its slope. x y

CORRESPONDING LENGTHS Without using the Corresponding Lengths Property on page 375, prove that the ratio of two corresponding angle bisectors in similar triangles is equal to the scale factor.

CHALLENGE Prove that if the lengths of two sides of a triangle are a and b respectively, then the lengths of the corresponding altitudes to those sides are in the ratio b }a .

In Exercises 4144, use the diagram. 2 6 4 8 1 Name three pairs of corresponding angles. (p. 147)

In Exercises 4144, use the diagram. 2 6 4 8 1 Name two pairs of alternate interior angles. (p. 147)

In Exercises 4144, use the diagram. 2 6 4 8 1 Name two pairs of alternate exterior angles. (p. 147)

In Exercises 4144, use the diagram. 2 6 4 8 1 Find m 1 1 m 7. (p. 154)

CONGRUENCE Explain why nABE > nCDE. (p. 240) A E D

Simplify the ratio. (p. 356) 4 }20

Simplify the ratio. (p. 356) 36 }18

Simplify the ratio. (p. 356) 21 : 63

Simplify the ratio. (p. 356) 42 : 28

VOCABULARY You plan to prove that nACB is similar to nPXQ by the SSS Similarity Theorem. Copy and complete the proportion that is needed to use this theorem: } AC ? 5 } ? XQ 5 } AB ? .

WRITING If you know two triangles are similar by the SAS Similarity Theorem, what additional piece(s) of information would you need to know to show that the triangles are congruent?

SSS SIMILARITY THEOREM Verify that nABC , nDEF. Find the scale factor of nABC to nDEF. nABC: BC 5 18, AB 5 15, AC 5 12 nDEF: EF 5 12, DE 5 10, DF 5 8

SSS SIMILARITY THEOREM Verify that nABC , nDEF. Find the scale factor of nABC to nDEF. nABC: AB 5 10, BC 5 16, CA 5 20 nDEF: DE 5 25, EF 5 40, FD 5 50

SSS SIMILARITY THEOREM Is either nJKL or nRST similar to nABC? B 8 C 12 11 6 7 K 7 L T 6 R S 4 3.5

SSS SIMILARITY THEOREM Is either n JKL or nRST similar to nABC? S T A C R 12 B 20 17.5 25 16 10.5 16

SAS SIMILARITY THEOREM Determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. W X Y 6 D

SAS SIMILARITY THEOREM Determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. A B 24 T S R K 18 10 8 1128 1128

ALGEBRA Find the value of n that makes nPQR , nXYZ when PQ 5 4, QR 5 5, XY 5 4(n 1 1), YZ 5 7n 2 1, and Q > Y. Include a sketch.

SHOWING SIMILARITY Show that the triangles are similar and write a similarity statement. Explain your reasoning. 16.5 5.5 18 15 5

SHOWING SIMILARITY Show that the triangles are similar and write a similarity statement. Explain your reasoning. A 27 18 14

SHOWING SIMILARITY Show that the triangles are similar and write a similarity statement. Explain your reasoning. X Z Y 30 21 35 478 47

ERROR ANALYSIS Describe and correct the students error in writing the similarity statement. n ABC , n PQR by SAS Similarity Theorem B R 18 24 20 15 86 86 C Q

MULTIPLE CHOICE In the diagram, } MN MR 5 } MP MQ . Which of the statements must be true? A 1 > 2 B }QR i }NP C 1 > 4 D nMNP , nMRQ P R N P M 2 3 1 4

DRAWING TRIANGLES Sketch the triangles using the given description. Explain whether the two triangles can be similar. In nXYZ, m X 5 668 and m Y 5 348. In nLMN, m M 5 348 and m N 5 808.

DRAWING TRIANGLES Sketch the triangles using the given description. Explain whether the two triangles can be similar. In nRST, RS 5 20, ST 5 32, and m S 5 168. In nFGH, GH 5 30, HF 5 48, and m H 5 248.

DRAWING TRIANGLES Sketch the triangles using the given description. Explain whether the two triangles can be similar. . The side lengths of nABC are 24, 8x, and 54, and the side lengths of nDEF are 15, 25, and 7x.

FINDING MEASURES In Exercises 1823, use the diagram to copy and complete the statements. 18 P S M N L 12 20 16 28 24 2 458 5 m NQP 5 ?

FINDING MEASURES In Exercises 1823, use the diagram to copy and complete the statements. 18 P S M N L 12 20 16 28 24 2 458 5 m QPN 5 ?

FINDING MEASURES In Exercises 1823, use the diagram to copy and complete the statements. 18 P S M N L 12 20 16 28 24 2 458 5 m PNQ 5 ?

FINDING MEASURES In Exercises 1823, use the diagram to copy and complete the statements. 18 P S M N L 12 20 16 28 24 2 458 5 RN 5 ?

FINDING MEASURES In Exercises 1823, use the diagram to copy and complete the statements. 18 P S M N L 12 20 16 28 24 2 458 5 PQ 5 ?

FINDING MEASURES In Exercises 1823, use the diagram to copy and complete the statements. 18 P S M N L 12 20 16 28 24 2 458 5 NM 5 ?

SIMILAR TRIANGLES In the diagram at the right, name the three pairs of triangles that are similar. L 12 20 16 28 24 2 458 5

CHALLENGE In the figure at the right, nABC , nVWX. 45 Y W B A 12 D C V 5 Find the scale factor of nVWX to nABC.

CHALLENGE In the figure at the right, nABC , nVWX. 45 Y W B A 12 D C V 5 Find the ratio of the area of nVWX to the area of nABC.

CHALLENGE In the figure at the right, nABC , nVWX. 45 Y W B A 12 D C V 5 Make a conjecture about the relationship between the scale factor in Exercise 25 and the ratio in Exercise 26. Justify your conjecture.

RACECAR NET Which postulate or theorem could you use to show that the three triangles that make up the racecar window net are similar? Explain. BG i }CF , }CF i }DE

STAINED GLASS Certain sections of stained glass are sold in triangular beveled pieces. Which of the three beveled pieces, if any, are similar? 3 in. 3 in. 5.25 in. 3 in. 7 in. 4 in. 4 in.

SHUFFLEBOARD In the portion of the shuffleboard court shown, } BC AC 5 } BD AE       /&& ! What additional piece of information do you need in order to show that nBCD , nACE using the SSS Similarity Theorem?

SHUFFLEBOARD In the portion of the shuffleboard court shown, } BC AC 5 } BD AE       /&& ! What additional piece of information do you need in order to show that nBCD , nACE using the SAS Similarity Theorem?

OPEN-ENDED MATH Use a diagram to show why there is no Side-Side-Angle Similarity Postulate.

MULTI-STEP PROBLEM Ruby is standing in her back yard and she decides to estimate the height of a tree. She stands so that the tip of her shadow coincides with the tip of the trees shadow, as shown. Ruby is 66 inches tall. The distance from the tree to Ruby is 95 feet and the distance between the tip of the shadows and Ruby is 7 feet. a. What postulate or theorem can you use to show that the triangles in the diagram are similar? b. About how tall is the tree, to the nearest foot? c. What If? Curtis is 75 inches tall. At a different time of day, he stands so that the tip of his shadow and the tip of the trees shadow coincide, as described above. His shadow is 6 feet long. How far is Curtis from the tree?

EXTENDED RESPONSE Suppose you are given two right triangles with one pair of corresponding legs and the pair of corresponding hypotenuses having the same length ratios. a. The lengths of the given pair of corresponding legs are 6 and 18, and the lengths of the hypotenuses are 10 and 30. Use the Pythagorean Theorem to solve for the lengths of the other pair of corresponding legs. Draw a diagram. b. Write the ratio of the lengths of the second pair of corresponding legs. c. Are these triangles similar? Does this suggest a Hypotenuse-Leg Similarity Theorem for right triangles?

PROOF Given that nABC is a right triangle and D, E, and F are midpoints, prove that m DEF 5 908. D F C E

WRITING Can two triangles have all pairs of corresponding angles in proportion? Explain.

PROVING THEOREM 6.3 Write a paragraph proof of the SAS Similarity Theorem. GIVEN c A > D, } AB DE 5 } AC DF PROVE c nABC , nDEF C E G H B

CHALLENGE A portion of a water slide in an amusement park is shown. Find the length of }EF . (Note: The posts form right angles with the ground.) FT FT ! $ % &

Find the slope of the line that passes through the given points. (p. 171) (0, 28), (4, 16)

Find the slope of the line that passes through the given points. (p. 171) (22, 29), (1, 23)

Find the slope of the line that passes through the given points. (p. 171) . (23, 9), (7, 2)

State the postulate or theorem you would use to prove the triangles congruent. Then write a congruence statement. (p. 249) R S P T

Find the value of x. DE is a midsegment of nABC. (p. 295) A E C D B 15

Find the value of x. GK GH 5 JK }FH (p. 364) F x H J K G

VOCABULARY State the Triangle Proportionality Theorem. Draw a diagram.

WRITING Compare the Midsegment Theorem (see page 295) and the Triangle Proportionality Theorem. How are they related?

FINDING THE LENGTH OF A SEGMENT Find the length of}AB 4 12 3 C B D A

FINDING THE LENGTH OF A SEGMENT Find the length of}AB 14 12 D C E B A

REASONING Use the given information to determine whether}KM i }JN . Explain your reasoning. 5 12 8 5 N J M K L

REASONING Use the given information to determine whether}KM i }JN . Explain your reasoning. 4 15 10 18 N J

REASONING Use the given information to determine whether}KM i }JN . Explain your reasoning. 25 22.5 20 18 J

MULTIPLE CHOICE For the figure at the right, which statement is not necessarily true? A PQ }QR 5 } UT TS B } TS UT 5 QR }PQ C QR }RS 5 TS }RS D PQ }PR 5 } UT US UT S P

ALGEBRA Find the value of the variable. 5 21 14

ALGEBRA Find the value of the variable. 8 4 6

ALGEBRA Find the value of the variable. z 3 4.5 1.5

ERROR ANALYSIS A student begins to solve for the length of }AD as shown. Describe and correct the students error. AB BC 5 AD }CD 10 }16 5} 20 2 x 20 20 10 16 C D A

MULTIPLE CHOICE Find the value of x. A 1 }2 B 1 C 2 D 3 2x 2x 1 1 6x 6 7.5 1

ALGEBRA Find the value of the variable. 16.5 11 29

ALGEBRA Find the value of the variable. q 36 28 1

FINDING SEGMENT LENGTHS Use the diagram to find the value of each variable. a c d e f 12 12.5 12.5 5 10 15

FINDING SEGMENT LENGTHS Use the diagram to find the value of each variable. b d c 6 6 3 6 9

RROR ANALYSIS A student claims that AB 5 AC using the method shown. Describe and correct the students error. By Theorem 6.7, BD }CD 5 } AB AC . Because BD 5 CD, it follows that AB 5 AC. B D A C

CONSTRUCTION Follow the instructions for constructing a line segment that is divided into four equal parts. a. Draw a line segment that is about 3 inches long, and label its endpoints A and B. Choose any point C not on }AB . Draw ] AC . b. Using any length, place the compass point at A and make an arc intersecting ] AC at D. Using the same compass setting, make additional arcs on ] AC . Label the points E, F, and G so that AD 5 DE 5 EF 5 FG. c. Draw }GB . Construct a line parallel to }GB through D. Continue constructing parallel lines and label the points as shown. Explain why AJ 5 JK 5 KL 5 LB. A B D E F G C AJ KL B D E F G C

CHALLENGE Given segments with lengths r, s, and t, construct a segment of length x, such that r }s 5 t }x r t s

CITY MAP On the map below, Idaho Avenue bisects the angle between University Avenue and Walter Street. To the nearest yard, what is the distance along University Avenue from 12th Street to Washington Street? 7ASHINGTON3T TH3T 7ALTER3T )DAHO!VE 5NIVERSITY!VE

PROVING THEOREM 6.4 Prove the Triangle Proportionality Theorem. GIVEN c }QS i }TU PROVE c QT }TR 5 } SU UR S T U

PROVING THEOREM 6.6 Use the diagram with the auxiliary line drawn to write a paragraph proof of Theorem 6.6. GIVEN c k1 i k2, k2 i k3 PROVE c } CB BA 5 } DE EF C D auxiliary line B E A F t1 t 2 k3

MULTI-STEP PROBLEM The real estate term lake frontage refers to the distance along the edge of a piece of property that touches a lake. a. Find the lake frontage (to the nearest tenth of a yard) for each lot shown. b. In general, the more lake frontage a lot has, the higher its selling price. Which of the lots should be listed for the highest price? c. Suppose that lot prices are in the same ratio as lake frontages. If the least expensive lot is $100,000, what are the prices of the other lots? Explain your reasoning. lake Lakeshore Drive Lot A Lot B Lot C 48 yd 55 yd 174 yd 61 yd

SHORT RESPONSE Sketch an isosceles triangle. Draw a ray that bisects the angle opposite the base. This ray divides the base into two segments. By Theorem 6.7, the ratio of the legs is proportional to the ratio of these two segments. Explain why this ratio is 1 : 1 for an isosceles triangle.

PLAN FOR PROOF Use the diagram given for the proof of Theorem 6.4 in Exercise 22 to write a plan for proving Theorem 6.5, the Triangle Proportionality Converse.

PROVING THEOREM 6.7 Use the diagram with the auxiliary lines drawn to write a paragraph proof of Theorem 6.7. GIVEN c YXW > WXZ PROVE c } YW WZ 5 } XY XZ auxiliary lines Z W Y X A

EXTENDED RESPONSE In perspective drawing, lines that are parallel in real life must meet at a vanishing point on the horizon. To make the train cars in the drawing appear equal in length, they are drawn so that the lines connecting the opposite corners of each car are parallel. a. Use the dimensions given and the red parallel lines to find the length of the bottom edge of the drawing of Car 2. b. What other set of parallel lines exist in the figure? Explain how these can be used to form a set of similar triangles. c. Find the length of the top edge of the drawing of Car 2.

CHALLENGE Prove Cevas Theorem: If P is any point inside nABC, then } AY YC p } CX XB p } BZ ZA 5 1. (Hint: Draw lines parallel to }BY through A and C. Apply Theorem 6.4 to nACM. Show that nAPN , nMPC, nCXM , nBXP, and nBZP , nAZN.) C B P Y N

Perform the following operations. Then simplify. (23) p 7 }2 (p. 869)

Perform the following operations. Then simplify. 4 }3 p 1 }2 (p. 869)

Perform the following operations. Then simplify. 51 1 }22 2 (p. 871)

Perform the following operations. Then simplify. 1 5 }42 3 (p. 871)

Describe the translation in words and write the coordinate rule for the translation. (p. 272) x y

Describe the translation in words and write the coordinate rule for the translation. (p. 272) x y 3

Describe the translation in words and write the coordinate rule for the translation. (p. 272) 3 2

VOCABULARY Copy and complete: In a dilation, the image is ? to the original figure.

WRITING Explain how to find the scale factor of a dilation. How do you know whether a dilation is an enlargement or a reduction?

DRAWING DILATIONS Draw a dilation of the polygon with the given vertices using the given scale factor k. A(22, 1), B(24, 1), C(22, 4); k 5 2

DRAWING DILATIONS Draw a dilation of the polygon with the given vertices using the given scale factor k. A(25, 5), B(25, 210), C(10, 0); k 5 3 }5

DRAWING DILATIONS Draw a dilation of the polygon with the given vertices using the given scale factor k. A(1, 1), B(6, 1), C(6, 3); k 5 1.5

DRAWING DILATIONS Draw a dilation of the polygon with the given vertices using the given scale factor k. A(2, 8), B(8, 8), C(16, 4); k 5 0.25

DRAWING DILATIONS Draw a dilation of the polygon with the given vertices using the given scale factor k. A(28, 0), B(0, 8), C(4, 0), D(0, 24); k 5 3 }8

DRAWING DILATIONS Draw a dilation of the polygon with the given vertices using the given scale factor k. A(0, 0), B(0, 3), C(2, 4), D(2, 21); k 5 13 }2

IDENTIFYING DILATIONS Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor. y B A

IDENTIFYING DILATIONS Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor. y A B

IDENTIFYING DILATIONS Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor. y B 1 A

IDENTIFYING DILATIONS Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor. B 3

MULTIPLE CHOICE You want to create a quadrilateral PQRS that is similar to quadrilateral JKLM. What are the coordinates of S? A (2, 4) B (4, 22) C (22, 24) D (24, 22) M K P R 2 L 5 24

ERROR ANALYSIS A student found the scale factor of the dilation from }AB to }CD to be 2 }5 . Describe and correct the students error. 1 1 A B C D } AB CD 5 2 }5

ERROR ANALYSIS A student says that the figure shown represents a dilation. What is wrong with this statement? x y

IDENTIFYING TRANSFORMATIONS Determine whether the transformation shown is a translation, reflection, rotation, or dilation. x y 1 1

IDENTIFYING TRANSFORMATIONS Determine whether the transformation shown is a translation, reflection, rotation, or dilation. x y 1

IDENTIFYING TRANSFORMATIONS Determine whether the transformation shown is a translation, reflection, rotation, or dilation. 1 3

FINDING SCALE FACTORS Find the scale factor of the dilation of Figure A to Figure B. Then give the unknown lengths of Figure A. y B A 3 m 6 8 10

FINDING SCALE FACTORS Find the scale factor of the dilation of Figure A to Figure B. Then give the unknown lengths of Figure A. y B A r 3 9 9 3 1

MULTIPLE CHOICE In the diagram shown, nABO is a dilation of nDEO. The length of a median of nABO is what percent of the length of the corresponding median of nDEO? A 50% B 75% C 1331 }3 % D 200% 1 E B A

SHORT RESPONSE Suppose you dilate a figure using a scale factor of 2. Then, you dilate the image using a scale factor of 1 }2 . Describe the size and shape of this new image.

CHALLENGE Describe the two transformations, the first followed by the second, that combined will transform nABC into nDEF. A(23, 3), B(23, 1), C(0, 1) D(6, 6), E(6, 2), F(0, 2)

CHALLENGE Describe the two transformations, the first followed by the second, that combined will transform nABC into nDEF. A(6, 0), B(9, 6), C(12, 6) D(0, 3), E(1, 5), F(2, 5)

BILLBOARD ADVERTISEMENT A billboard advertising agency requires each advertisement to be drawn so that it fits in a 12-inch by 6-inch rectangle. The agency uses a scale factor of 24 to enlarge the advertisement to create the billboard. What are the dimensions of a billboard, in feet?

POTTERY Your pottery is used on a poster for a student art show. You want to make postcards using the same image. On the poster, the image is 8 inches in width and 6 inches in height. If the image on the postcard can be 5 inches wide, what scale should you use for the image on the postcard?

SHADOWS You and your friend are walking at night. You point a flashlight at your friend, and your friends shadow is cast on the building behind him. The shadow is an enlargement, and is 15 feet tall. Your friend is 6 feet tall. What is the scale factor of the enlargement?

OPEN-ENDED MATH Describe how you can use dilations to create the figure shown below. x y 9 5

MULTI-STEP PROBLEM nABC has vertices A(3, 23), B(3, 6), and C(15, 6). a. Draw a dilation of nABC using a scale factor of 2 }3 . b. Find the ratio of the perimeter of the image to the perimeter of the original figure. How does this ratio compare to the scale factor? c. Find the ratio of the area of the image to the area of the original figure. How does this ratio compare to the scale factor?

EXTENDED RESPONSE Look at the coordinate notation for a dilation on page 409. Suppose the definition of dilation allowed k < 0. a. Describe the dilation if 21 < k < 0. b. Describe the dilation if k < 21. c. Use a rotation to describe a dilation with k 5 21.

SHORT RESPONSE Explain how you can use dilations to make a perspective drawing with the center of dilation as a vanishing point. Draw a diagram. x y O

MIDPOINTS Let }XY be a dilation of }PQ with scale factor k. Show that the image of the midpoint of }PQ is the midpoint of }XY .

REASONING In Exercise 32, show that }XY i }PQ .

CHALLENGE A rectangle has vertices A(0, 0), B(0, 6), C(9, 6), and D(9, 0). Explain how to dilate the rectangle to produce an image whose area is twice the area of the original rectangle. Make a conjecture about how to dilate any polygon to produce an image whose area is n times the area of the original polygon.

Simplify the expression. (p. 873) (3x 1 2)2 1 (x 2 5)2

Simplify the expression. (p. 873) . 41 1 }2 ab2 1 (b 2 a) 2

Simplify the expression. (p. 873) (a 1 b) 2 2 (a 2 b) 2

Find the distance between each pair of points. (p. 15) (0, 5) and (4, 3)

Find the distance between each pair of points. (p. 15) (23, 0) and (2, 4)

Find the distance between each pair of points. (p. 15) (22, 24) and (3, 22)

Find the value(s) of the variable(s). Area 5 6 in.2 (p. 49) 3 in. h in. Y

Find the value(s) of the variable(s). n ABC > n DCB (p. 256) D y 1 7 3(x 1 1

Find the value(s) of the variable(s). n PQR is isosceles. (p. 303) 8 x 8 708 x 8 y 8 Q R P

Copy and complete the statement. A ? is a transformation in which the original figure and its image are similar.

Copy and complete the statement. If nPQR , nXYZ, then PQ }XY 5 ? }YZ 5 } ? ?

Copy and complete the statement. WRITING Describe the relationship between a ratio and a proportion. Give an example of each.

The length of a rectangle is 20 meters and the width is 15 meters. Find the ratio of the width to the length of the rectangle. Then simplify the ratio.

The measures of the angles in nUVW are in the extended ratio of 1 : 1 : 2. Find the measures of the angles.

Find the geometric mean of 8 and 12.

Use the diagram and the given information to find the unknown length. Given } RN RP 5 QM }QL , find RP. 6 10 M N P

Use the diagram and the given information to find the unknown length. Given } CD DB 5 } CE EA, find CD. E A 5 4 10 B

In Exercises 9 and 10, determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor. 9 12 8 H D G C A

In Exercises 9 and 10, determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor. 25 10 15 Z 8 6 P Y R

POSTERS Two similar posters have a scale factor of 4 : 5. The large posters perimeter is 85 inches. Find the small posters perimeter.

Use the AA Similarity Postulate to show that the triangles are similar. S R 358 358 P

Use the AA Similarity Postulate to show that the triangles are similar. C F D 608 308

CELL TOWER A cellular telephone tower casts a shadow that is 72 feet long, while a tree nearby that is 27 feet tall casts a shadow that is 6 feet long. How tall is the tower?

Use the SSS Similarity Theorem or SAS Similarity Theorem to show that the triangles are similar. 8 4 7

Use the SSS Similarity Theorem or SAS Similarity Theorem to show that the triangles are similar. 15 10 S R U T P

Use the given information to determine whether}AB i }CD . 8 20 16 10 C

Use the given information to determine whether}AB i }CD . 3.5 22.5 12 20 C

Draw a dilation of the polygon with the given vertices using the given scale factor k. T(0, 8), U(6, 0), V(0, 0); k 5 3 }2

Draw a dilation of the polygon with the given vertices using the given scale factor k. A(6, 0), B(3, 9), C(0, 0), D(3, 1); k 5 4

Draw a dilation of the polygon with the given vertices using the given scale factor k. P(8, 2), Q(4, 0), R(3, 1), S(6, 4); k 5 0.5