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: $A = 60 °, B = 15 °, c = 6$ .

123

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.

135

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?

158

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

215

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

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

159

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

Three sides are known.

163

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

164

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

144

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

229

Solve each triangle ABC.

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143

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

C = 41° 20', b = 25.9 m, c = 38.4 m

226

Solve each triangle ABC.

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154

Solve each triangle ABC.

A = 68.41°, B = 54.23°, a = 12.75 ft

166

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

B = 74.3°, a = 859 m, b = 783 m

215

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

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

187

Solve each triangle ABC.

B = 38° 40', a = 19.7 cm, C = 91° 40'

187

Solve each triangle ABC that exists.

A = 42.5°, a = 15.6 ft, b = 8.14 ft

229

Solve each triangle ABC.

A = 39.70°, C = 30.35°, b = 39.74 m

157

Solve each triangle ABC.

B = 42.88°, C = 102.40°, b = 3974 ft

148

Solve each triangle ABC that exists.

B = 72.2°, b = 78.3 m, c = 145 m

239

Solve each triangle ABC that exists.

A = 38° 40', a = 9.72 m, b = 11.8 m

148

Solve each triangle ABC that exists.

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

167

Solve each triangle ABC.

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

146

Solve each triangle ABC that exists.

B = 39.68°, a = 29.81 m, b = 23.76 m

213

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

200

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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161

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?

391

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.

221

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.

430

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?

180

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°

183

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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215

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

a = 50, b = 26, A = 95°

292

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

156

Find the area of each triangle ABC.

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

177

Find the area of each triangle ABC.

A = 59.80°, b = 15.00 cm, C = 53.10°

144

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?

147

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?

175

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

a = 31, b = 26, B = 48°

209

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

c = 50, b = 61, C = 58°

272

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

290

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

140

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

216

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

244

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

368

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°

158

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°

178

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°

253

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°

190

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°

227

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°

165

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°

195

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°

457

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

248

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

288

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

266

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

351

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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235

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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147

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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142

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.

217

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?

179

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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220

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

15

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

17