7. Non-Right Triangles

Multiple Choice

Use the Law of Sines to find the angle $B$ to the nearest tenth of a degree.

Classify the triangle, then solve: $A = 60 °, B = 15 °, c = 6$ .

124

Multiple Choice

An engineer wants to measure the distance to cross a river. If $B=30\degree$ , $a=300$ $ft$ , $C=100\degree$ find the shortest distance (in $ft$ ) you’d have to travel to cross the river.

137

Apply the law of sines to the following: a = √5, c = 2√5, A = 30°. What is the value of sin C? What is the measure of C? Based on its angle measures, what kind of triangle is triangle ABC?

159

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1

216

Fill in the blank(s) to correctly complete each sentence.

A triangle that is not a right triangle is a(n) _________ triangle.

160

Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines.

Three sides are known.

164

Which one of the following sets of data does not determine a unique triangle?

a. A = 50°, b = 21, a = 19

b. A = 45°, b = 10, a = 12

c. A = 130°, b = 4, a = 7

d. A = 30°, b = 8, a = 4

166

Use the law of sines to find the indicated part of each triangle ABC.

Find b if C = 74.2°, c = 96.3 m, B = 39.5

145

Find each angle B. Do not use a calculator.

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

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Find the unknown angles in triangle ABC for each triangle that exists.

A = 29.7°, b = 41.5 ft, a = 27.2 ft

230

Solve each triangle ABC.

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144

Find the unknown angles in triangle ABC for each triangle that exists.

C = 41° 20', b = 25.9 m, c = 38.4 m

228

Solve each triangle ABC.

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155

Solve each triangle ABC.

A = 68.41°, B = 54.23°, a = 12.75 ft

167

Find the unknown angles in triangle ABC for each triangle that exists.

B = 74.3°, a = 859 m, b = 783 m

217

Find the unknown angles in triangle ABC for each triangle that exists.

A = 142.13°, b = 5.432 ft, a = 7.297 ft

188

Solve each triangle ABC.

B = 38° 40', a = 19.7 cm, C = 91° 40'

188

Solve each triangle ABC that exists.

A = 42.5°, a = 15.6 ft, b = 8.14 ft

231

Solve each triangle ABC.

A = 39.70°, C = 30.35°, b = 39.74 m

158

Solve each triangle ABC.

B = 42.88°, C = 102.40°, b = 3974 ft

149

Solve each triangle ABC that exists.

B = 72.2°, b = 78.3 m, c = 145 m

242

Solve each triangle ABC that exists.

A = 38° 40', a = 9.72 m, b = 11.8 m

150

Solve each triangle ABC that exists.

A = 96.80°, b = 3.589 ft, a = 5.818 ft

168

Solve each triangle ABC.

C = 79° 18', c = 39.81 mm, A = 32° 57'

147

Solve each triangle ABC that exists.

B = 39.68°, a = 29.81 m, b = 23.76 m

214

Use the law of sines to find the indicated part of each triangle ABC.

Find B if C = 51.3°, c = 68.3 m, b = 58.2 m

202

To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.

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306

To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.

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162

A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?

394

Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.

222

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

432

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

181

Use the law of sines to find the indicated part of each triangle ABC.

Find b if a = 165 m, A = 100.2°, B = 25.0°

184

Find the area of each triangle using the formula 𝓐 = ½ bh, and then verify that the formula 𝓐 = ½ ab sin C gives the same result.

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216

Determine the number of triangles ABC possible with the given parts.

a = 50, b = 26, A = 95°

293

Find the area of each triangle ABC.

A = 42.5°, b = 13.6 m, c = 10.1 m

Find the area of each triangle ABC.

B = 124.5°, a = 30.4 cm, c = 28.4 cm

157

Find the area of each triangle ABC.

A = 56.80°, b = 32.67 in., c = 52.89 in.

178

Find the area of each triangle ABC.

A = 59.80°, b = 15.00 cm, C = 53.10°

145

A painter is going to apply paint to a triangular metal plate on a new building. Two sides measure 16.1 m and 15.2 m, and the angle between the sides is 125°. What is the area of the surface to be painted?

148

A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 m and 21.3 m, and the angle between them is 42.2°. What is the area of the triangular lot?

176

Determine the number of triangles ABC possible with the given parts.

a = 31, b = 26, B = 48°

214

Determine the number of triangles ABC possible with the given parts.

c = 50, b = 61, C = 58°

273

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. A = 56°, C = 24°, a = 22

291

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 37°, a = 12.4, b = 8.7

142

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. A = 85°, B = 35°, c = 30

217

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. B = 5°, C = 125°, b = 200

245

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. B = 80°, C = 10°, a = 8

369

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 30, b = 20, A = 50°

159

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 57.5, c = 49.8, A = 136°

179

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 6.1, b = 4, A = 162°

255

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 10, b = 30, A = 150°

191

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 30, b = 40, A = 20°

228

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 7, b = 28, A = 12°

166

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 95, c = 125, A = 49°

196

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 1.4, b = 2.9, A = 142°

458

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. A = 22°, b = 20 feet, c = 50 feet

249

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. B = 125°, a = 8 yards, c = 5 yards

289

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. C = 102°, a = 16 meters, b = 20 meters

267

In Exercises 43–44, use the given measurements to solve the following triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. a = 400, b = 300

353

In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?

a. two triangles

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236

In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?

b. exactly one triangle

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148

In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?

c. no triangle

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143

Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.

218

Apply the law of sines to the following:

A = 104°, a = 26.8, b = 31.3.

What happens when we try to find the measure of angle B using a calculator?

180

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.

(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon?

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221

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

16

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

18