Chapter 14

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle M N P, m=16, n=15, p=8\)

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ P Q=12, P R=15, \mathrm{m} \angle R=100 $$

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle D E F, d=5.83, e=5.83, \mathrm{m} \angle F=48\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=12, \theta=270^{\circ}\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle A B C, a=10, b=12,\) and \(\mathrm{m} \angle B=30\)

In \(\triangle A B C, \mathrm{m} \angle A=102, \mathrm{m} \angle B=34,\) and \(a=25.8 .\) Find \(c\) to the nearest tenth.

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle A B C, a=5, b=12, c=13\)

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ B C=12, A C=12 \sqrt{2}, \mathrm{m} \angle B=135 $$

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle P Q R, p=212, q=287, \mathrm{m} \angle R=124\)

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle R S T, R S=9, S T=9 \sqrt{3},\) and \(\mathrm{m} \angle S=\frac{5 \pi}{6}\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=\sqrt{2}, \theta=225^{\circ}\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle A B C, b=25, \mathrm{m} \angle B=45,\) and \(\mathrm{m} \angle C=60\)

In \(\triangle P Q R, \mathrm{m} \angle P=125, \mathrm{m} \angle Q=14,\) and \(p=122 .\) Find \(r\) to the nearest integer.

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 12,20,22 $$

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ R S=3 \sqrt{3}, S T=3, \mathrm{m} \angle T=60 $$

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle R S T, t=15.7, s=15.7, \mathrm{m} \angle R=98\)

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle A B C, A B=2 \sqrt{2}, B C=4,\) and \(\mathrm{m} \angle B=\frac{3 \pi}{4}\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=\sqrt{3}, \theta=300^{\circ}\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle A B C, a=8, \mathrm{m} \angle B=35,\) and \(\mathrm{m} \angle C=55\)

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 9,10,15 $$

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ a=8, b=10, \mathrm{m} \angle A=45 $$

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle D E F, e=336, f=257, \mathrm{m} \angle D=122\)

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle A B C, b=12.4, c=8.70,\) and \(\mathrm{m} \angle A=23\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=2, \theta=-60^{\circ}\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle D E F, d=72, e=48,\) and \(\mathrm{m} \angle F=110\)

In \(\triangle C D E, \mathrm{m} \angle D=125, \mathrm{m} \angle E=28,\) and \(d=12.5 .\) Find \(c\) to the nearest hundredth.

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 30,35,45 $$

A ladder that is 15 feet long is placed so that it reaches from level ground to the top of a vertical wall that is 13 feet high. a. Use the Law of Sines to find the angle that the ladder makes with the ground to the nearest hundredth. b. Is more than one position of the ladder possible? Explain your answer.

Find the exact value of the area of an equilateral triangle if the length of one side is 40 meters.

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle P Q R, p=126, q=214,\) and \(\mathrm{m} \angle R=42\)

The coordinates of a point are given. a. Find the distance of the point from the origin. Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains the given point. \((6,8)\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle P Q R, p=12, \mathrm{m} \angle Q=80,\) and \(\mathrm{m} \angle R=30\)

The base of an isosceles triangle measures 14.5 centimeters and the vertex angle measures 110 degrees. a. Find the measure of one of the congruent sides of the triangle to the nearest hundredth. b. Find the perimeter of the triangle to the nearest tenth.

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 11,11,15 $$

Max has a triangular garden. He measured two sides of the garden and the angle opposite one of these sides. He said that the two sides measured 5 feet and 8 feet and that the angle opposite the 8 -foot side measured 75 degrees. Can a garden exist with these measurements? Could there be two gardens of different shapes with these measurements? Write the angle measures and lengths of the sides of the garden(s) if any.

Find the exact value of the area of an isosceles triangle if the measure of a leg is 12 centimeters and the measure of the vertex angle is 45 degrees.

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle D E F, d=3.25, e=5.62,\) and \(\mathrm{m} \angle F=58\)

The coordinates of a point are given. a. Find the distance of the point from the origin. Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains the given point. \((-5,12)\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle R S T, r=38, s=28,\) and \(t=18\)

The length of one of the equal sides of an isosceles triangle measures 25.8 inches and each base angle measures 53 degrees. a. Find the measure of the base of the triangle to the nearest tenth. b. Find the perimeter of the triangle to the nearest inch.

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 32,40,38 $$

Emily wants to draw a parallelogram with the measure of one side 12 centimeters, the measure of one diagonal 10 centimeters and the measure of one angle 120 degrees. Is this possible? Explain why or why not.

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle A B C, a=62.5, b=44.7,\) and \(\mathrm{m} \angle C=133\)

Find the area of a parallelogram if the measures of two adjacent sides are 40 feet and 24 feet and the measure of one angle of the parallelogram is 30 degrees.

The coordinates of a point are given. a. Find the distance of the point from the origin. Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains the given point. \((0,7)\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle A B C, a=22, b=18,\) and \(\mathrm{m} \angle C=130\)

Use the Law of Sines to show that if \(\angle C\) of \(\triangle A B C\) is a right angle, \(\sin A=\frac{a}{c}\) .

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 7,24,25 $$

Ross said that when he jogs, his path forms a triangle. Two sides of the triangle are 2.0 kilo- meters and 2.5 kilometers in length and the angle opposite the shorter side measures 45 degrees. Rosa said that when she jogs, her path also forms a triangle with two sides of length 2.0 kilometers and 2.5 kilometers and an angle of 45 degrees opposite the shorter side. Rosa said that her route is longer than the route Ross follows. Is this possible? Explain your answer.

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle R S T, R S=0.375, S T=1.29,\) and \(\mathrm{m} \angle S=167\)