Distance between a Ship and a Submarine From an airplane

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. a, b, and C

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. A, C, and c

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. a, b, and A

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. a, B, and C

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. A, B, and c

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. a, c, and A

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. a, b, and c

Concept Check Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. b, c, and A

Find the length of the remaining side of each triangle. Do not use a calculator.

Find the length of the remaining side of each triangle. Do not use a calculator.

Find the measure of u in each triangle. Do not use a calculator.

Find the measure of u in each triangle. Do not use a calculator.

Solve each triangle. Approximate values to the nearest tenth.

Solve each triangle. Approximate values to the nearest tenth.

Solve each triangle. Approximate values to the nearest tenth.

Solve each triangle. Approximate values to the nearest tenth.

Solve each triangle. Approximate values to the nearest tenth.

Solve each triangle. Approximate values to the nearest tenth.

Solve each triangle. See Examples 2 and 3. A = 41.4_, b = 2.78 yd, c = 3.92 yd 2

Solve each triangle. See Examples 2 and 3. C = 28.3_, b = 5.71 in., a = 4.21 in. 2

Solve each triangle. See Examples 2 and 3. C = 45.6_, b = 8.94 m, a = 7.23 m 2

Solve each triangle. See Examples 2 and 3. A = 67.3_, b = 37.9 km, c = 40.8 km 2

Solve each triangle. See Examples 2 and 3. a = 9.3 cm, b = 5.7 cm, c = 8.2 cm 2

Solve each triangle. See Examples 2 and 3. a = 28 ft, b = 47 ft, c = 58 ft 2

Solve each triangle. See Examples 2 and 3. a = 42.9 m, b = 37.6 m, c = 62.7 m 2

Solve each triangle. See Examples 2 and 3. a = 189 yd, b = 214 yd, c = 325 yd 2

Solve each triangle. See Examples 2 and 3. a = 965 ft, b = 876 ft, c = 1240 ft 2

Solve each triangle. See Examples 2 and 3. a = 324 m, b = 421 m, c = 298 m 2

Solve each triangle. See Examples 2 and 3. A = 80_ 40_, b = 143 cm, c = 89.6 cm 3

Solve each triangle. See Examples 2 and 3. C = 72_ 40_, a = 327 ft, b = 251 ft 3

Solve each triangle. See Examples 2 and 3. B = 74.8_, a = 8.92 in., c = 6.43 in. 3

Solve each triangle. See Examples 2 and 3. C = 59.7_, a = 3.73 mi, b = 4.70 mi 3

Solve each triangle. See Examples 2 and 3. A = 112.8_, b = 6.28 m, c = 12.2 m 3

Solve each triangle. See Examples 2 and 3. B = 168.2_, a = 15.1 cm, c = 19.2 cm 3

Solve each triangle. See Examples 2 and 3. a = 3.0 ft, b = 5.0 ft, c = 6.0 ft 3

Solve each triangle. See Examples 2 and 3. a = 4.0 ft, b = 5.0 ft, c = 8.0 ft 3

Refer to Figure 12. If you attempt to find any angle of a triangle with the values a = 3, b = 4, and c = 10 by using the law of cosines, what happens? 3

The shortest distance between two points is a straight line. Explain how this is related to the geometric property that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Solve each problem. See Examples 14. 3

Distance across a River Points A and B are on opposite sides of False River. From a third point, C, the angle between the lines of sight to A and B is 46.3. If AC is 350 m long and BC is 286 m long, find AB. 4

Distance across a Ravine Points X and Y are on opposite sides of a ravine. From a third point Z, the angle between the lines of sight to X and Y is 37.7. If XZ is 153 m long and YZ is 103 m long, find XY. 4

Angle in a Parallelogram A parallelogram has sides of length 25.9 cm and 32.5 cm. The longer diagonal has length 57.8 cm. Find the measure of the angle opposite the longer diagonal. 4

Diagonals of a Parallelogram The sides of a parallelogram are 4.0 cm and 6.0 cm. One angle is 58 while another is 122. Find the lengths of the diagonals of the parallelogram. 4

Flight Distance Airports A and B are 450 km apart, on an east-west line. Tom flies in a northeast direction from A to airport C. From C he flies 359 km on a bearing of 128_ 40_ to B. How far is C from A? 4

Distance Traveled by a Plane An airplane flies 180 mi from point X at a bearing of 125_, and then turns and flies at a bearing of 230_ for 100 mi. How far is the plane from point X? 4

Distance between Ends of the Vietnam Memorial The Vietnam Veterans Memorial in Washington, D.C., is V-shaped with equal sides of length 246.75 ft. The angle between these sides measures 125_ 12_ . Find the distance between the ends of the two sides. (Source: Pamphlet obtained at Vietnam Veterans Memorial.) 246.75 ft 246.75 ft 125 12_ 4

Distance between Two Ships Two ships leave a harbor together, traveling on courses that have an angle of 135_ 40_ between them. If each travels 402 mi, how far apart are they? 135 40_ Harbor 402 mi 402 mi 4

Distance between a Ship and a Rock A ship is sailing east. At one point, the bearing of a submerged rock is 45_ 20_. After the ship has sailed 15.2 mi, the bearing of the rock has become 308_ 40_. Find the distance of the ship from the rock at the latter point. 4

Distance between a Ship and a Submarine From an airplane flying over the ocean, the angle of depression to a submarine lying under the surface is 24_ 10_. At the same moment, the angle of depression from the airplane to a battleship is 17_ 30_. See the figure. The distance from the airplane to the battleship is 5120 ft. Find the distance between the battleship and the submarine. (Assume the airplane, submarine, and battleship are in a vertical plane.) Submarine 24 10_ 5120 ft 17 30_ Battleship 4

Truss Construction A triangular truss is shown in the figure. Find angle u. 5

Truss Construction Find angle b in the truss shown in the figure. 5

Distance between a Beam and Cables A weight is supported by cables attached to both ends of a balance beam, as shown in the figure. What angles are formed between the beam and the cables? 5

Distance between Points on a Crane A crane with a counterweight is shown in the figure. Find the horizontal distance between points A and B to the nearest foot. _ _ 20 ft 16 ft 13 ft 90 ft ? ? 45 ft 60 ft 10 ft 10 ft C A B 128 5

Distance on a Baseball Diamond A baseball diamond is a square, 90.0 ft on a side, with home plate and the three bases as vertices. The pitchers position is 60.5 ft from home plate. Find the distance from the pitchers position to each of the bases. 3rd base Home plate 1st base 90.0 ft 2nd base 60.5 ft 5

Distance on a Softball Diamond A softball diamond is a square, 60.0 ft on a side, with home plate and the three bases as vertices. The pitchers position is 46.0 ft from home plate. Find the distance from the pitchers position to each of the bases. 5

Distance between a Ship and a Point Starting at point A, a ship sails 18.5 km on a bearing of 189, then turns and sails 47.8 km on a bearing of 317. Find the distance of the ship from point A. 5

Distance between Two Factories Two factories blow their whistles at exactly 5:00. A man hears the two blasts at 3 sec and 6 sec after 5:00, respectively. The angle between his lines of sight to the two factories is 42.2. If sound travels 344 m per sec, how far apart are the factories? 5

Measurement Using Triangulation Surveyors are often confronted with obstacles, such as trees, when measuring the boundary of a lot. One technique used to obtain an accurate measurement is the so-called triangulation method. In this technique, a triangle is constructed around the obstacle and one angle and two sides of the triangle are measured. Use this technique to find the length of the property line (the straight line between the two markers) in the figure. (Source: Kavanagh, B., Surveying Principles and Applications, Sixth Edition, Prentice-Hall.) 14.0 ft 70.0 13.0 ft Not to scale Marker Marker 18.0 ft 14.0 ft 5

Path of a Ship A ship sailing due east in the North Atlantic has been warned to change course to avoid icebergs. The captain turns and sails on a bearing of 62, then changes course again to a bearing of 115 until the ship reaches its original course. See the figure. How much farther did the ship have to travel to avoid the icebergs? Icebergs 50 mi E 5

Length of a Tunnel To measure the distance through a mountain for a proposed tunnel, a point C is chosen that can be reached from each end of the tunnel. See the figure. If AC = 3800 m, BC = 2900 m, and angle C = 110_ , find the length of the tunnel. A B 110 3800 m 2900 m C Tunnel 6

Distance between an Airplane and a Mountain A person in a plane flying straight north observes a mountain at a bearing of 24.1. At that time, the plane is 7.92 km from the mountain. A short time later, the bearing to the mountain becomes 32.7. How far is the airplane from the mountain when the second bearing is taken? 6

Find the measure of each angle u to two decimal places.

Find the measure of each angle u to two decimal places.

Find the exact area of each triangle using the formula = 12 bh, and then verify that Herons formula gives the same result.

Find the exact area of each triangle using the formula = 12 bh, and then verify that Herons formula gives the same result.

Find the area of each triangle ABC. See Example 5. a = 12 m, b = 16 m, c = 25 m 6

Find the area of each triangle ABC. See Example 5. a = 22 in., b = 45 in., c = 31 in. 6

Find the area of each triangle ABC. See Example 5. a = 154 cm, b = 179 cm, c = 183 cm 6

Find the area of each triangle ABC. See Example 5. a = 25.4 yd, b = 38.2 yd, c = 19.8 yd 6

Find the area of each triangle ABC. See Example 5. a = 76.3 ft, b = 109 ft, c = 98.8 ft 7

Find the area of each triangle ABC. See Example 5. a = 15.89 m, b = 21.74 m, c = 10.92 m 7

Perfect Triangles A perfect triangle is a triangle whose sides have whole number lengths and whose area is numerically equal to its perimeter. Show that the triangle with sides of length 9, 10, and 17 is perfect. 7

Heron Triangles A Heron triangle is a triangle having integer sides and area. Show that each of the following is a Heron triangle. (a) a = 11, b = 13, c = 20 (b) a = 13, b = 14, c = 15 (c) a = 7, b = 15, c = 20 (d) a = 9, b = 10, c = 17 7

Area of the Bermuda Triangle Find the area of the Bermuda Triangle if the sides of the triangle have approximate lengths 850 mi, 925 mi, and 1300 mi. 7

Required Amount of Paint A painter needs to cover a triangular region 75 m by 68 m by 85 m. A can of paint covers 75 m2 of area. How many cans (to the next higher number of cans) will be needed? 7

Consider triangle ABC shown here. (a) Use the law of sines to find candidates for the value of angle C. Round angle measures to the nearest tenth of a degree. (b) Rework part (a) using the law of cosines. (c) Why is the law of cosines a better method in this case? 7

Show that the measure of angle A is twice the measure of angle B. (Hint: Use the law of cosines to find cos A and cos B, and then show that cos A = 2 cos2 B - 1.) 7

Draw a triangle with vertices A12, 52, B1-1, 32, and C14, 02, and use the distance formula to find the lengths of the sides a, b, and c. 7

Find the area of triangle ABC using formula (b). (First use the law of cosines to find the measure of an angle.) 7

Find the area of triangle ABC using formula (c)that is, Herons formula. 8

Find the area of triangle ABC using new formula (d).