Chapter 14

A field is bordered by two pairs of parallel roads so that the shape of the field is a parallelogram. The lengths of two adjacent sides of the field are 2 kilometers and 3 kilometers, and the length of the shorter diagonal of the field is 3 kilometers. a. Find the cosine of the acute angle of the parallelogram. b. Find the exact value of the sine of the acute angle of the parallelogram. c. Find the exact value of the area of the field. d. Find the area of the field to the nearest integer.

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. \((12,-9)\)

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 \(\triangle P Q R, p=12, q=16,\) and \(r=20\)

A telephone pole on a hillside makes an angle of 78 degrees with the upward slope. A wire from the top of the pole to a point up the hill is 12.0 feet long and makes an angle of 15 degrees with the pole. a. Find, to the nearest hundredth, the distance from the foot of the pole to the point at which the wire is fastened to the ground. b. Use the answer to part a to find, to the nearest tenth, the height of the pole.

Two lighthouses are 12 miles apart along a straight shore. A ship is 15 miles from one light-house and 20 miles from the other. Find, to the nearest degree, the measure of the angle between the lines of sight from the ship to each lighthouse.

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle A B C, A B=2.35, B C=6.24,\) and \(\mathrm{m} \angle B=115\)

The roof of a shed consists of four congruent isosceles triangles. The length of each equal side of one triangular section is 22.0 feet and the measure of the vertex angle of each triangle is \(75^{\circ} .\) Find, to the nearest square foot, the area of one triangular section of the roof.

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. \((15,0)\)

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 \(\triangle D E F, d=36, e=72,\) and \(\mathrm{m} \angle D=30\)

A tree is braced by wires 4.2 feet and 4.7 feet long that are fastened to the tree at the same point and to the ground at points 7.8 feet apart. Find, to the nearest degree, the measure of the angle between the wires at the tree.

A garden is in the shape of an isosceles trapezoid. The lengths of the parallel sides of the garden are 30 feet and 20 feet, and the length of each of the other two sides is 10 feet. If a base angle of the trapezoid measures \(60^{\circ},\) find the exact area of the garden.

Ann and Bill Bekebrede follow a familiar triangular path when they take a walk. They walk mfrom home for 0.52 mile along a straight road, turn at an angle of \(95^{\circ},\) walk for another 0.46 mile, and then return home. a. Find, to the nearest hundredth of a mile, the length of the last portion of their walk. b. Find, to the nearest hundredth of a mile, the total distance that they walk.

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. \((-8,-12)\)

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 \(\triangle R S T, r=15, s=18,\) and \(\mathrm{m} \angle T=90\)

A kite is in the shape of a quadrilateral with two pair of congruent adjacent sides. The lengths of two sides are 20.0 inches and the lengths of the other two sides are 35.0 inches. The two shorter sides meet at an angle of \(115^{\circ} .\) a. Find the length of the diagonal between the points at which the unequal sides meet. Write the length to the nearest tenth of an inch. b. Using the answer to part a, find, to the nearest degree, the measure of the angle at which the two longer sides meet.

When two forces act on an object, the resultant force is the single force that would have produced the same result. When the magnitudes of the two forces are represented by the lengths of two sides of a parallelogram, the resultant can be represented by the length of the diagonal of the parallelogram. If forces of 12 pounds and 18 pounds act at an angle of \(75^{\circ}\) , what is the magnitude of the resultant force to the nearest hundredth pound?

In \(\triangle A B C, \mathrm{m} \angle B=30\) and in \(\triangle D E F, \mathrm{m} \angle E=150 .\) Show that if \(A B=D E\) and \(B C=E F,\) the areas of the two triangles are equal.

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. \((24,7)\)

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 \(\triangle A B C, a=15, b=25,\) and \(c=12\)

A distress signal from a ship, \(S,\) is received by two coast guard stations located 3.8 miles apart along a straight coastline. From station \(A,\) the signal makes an angle of \(48^{\circ}\) with the coastline and from station \(B\) the signal makes an angle of \(67^{\circ}\) with the coastline. Find, to the nearest tenth of a mile, the distance from the ship to the nearer station.

A field is in the shape of a parallelogram. The lengths of two adjacent sides are 48 meters and 65 meters. The measure of one angle of the parallelogram is \(100^{\circ} .\) a. Find, to the nearest meter, the length of the longer diagonal. b. Find, to the nearest meter, the length of the shorter diagonal.

Aaron wants to draw \(\triangle A B C\) with \(A B=15\) inches, \(B C=8\) inches, and an area of 40 square inches. a. What must be the sine of \(\angle B ?\) b. Find, to the nearest tenth of a degree, the measure of \(\angle B .\) c. Is it possible for Aaron to draw two triangles that are not congruent to each other that satisfy the given conditions? Explain.

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. \((6,-10)\)

Two sides of a triangular lot form angles that measure \(29.1^{\circ}\) and \(33.7^{\circ}\) with the third side, which is 487 feet long. To the nearest dollar, how much will it cost to fence the lot if the fencing costs \(\$ 5.59\) per foot?

A walking trail is laid out in the shape of a triangle. The lengths of the three paths that make up the trail are \(2,500\) meters, \(2,000\) meters, and \(1,800\) meters. Determine, to the nearest degree, the measure of the greatest angle of the trail.

A pole is braced by two wires that extend from the top of the pole to the ground. The lengths of the wires are 16 feet and 18 feet and the measure of the angle between the wires is \(110^{\circ} .\) Find, to the nearest foot, the distance between the points at which the wires are fastened to the ground.

Let \(A B C D\) be a parallelogram with \(A B=c, B C=a,\) and \(\mathrm{m} \angle B=\theta .\) a. Write a formula for the area of parallelogram \(A B C D\) in terms of \(c, a,\) and \(\theta\) . b. For what value of \(\theta\) does parallelogram \(A B C D\) have the greatest area?

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. \((-8,8)\)

A small park is in the shape of an isosceles trapezoid. The length of the longer of the parallel sides is 3.2 kilometers and the length of an adjacent side is 2.4 kilometers. A path from one corner of the park to an opposite corner is 3.6 kilometers long. a. Find, to the nearest tenth, the measure of each angle between adjacent sides of the park. b. Find, to the nearest tenth, the measure of each angle between the path and a side of the park. c. Find, to the nearest tenth, the length of the shorter of the parallel sides.

Use the formula cos \(C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}\) to show that the measure of each angle of an equilateral triangle is \(60^{\circ} .\)

Two points \(A\) and \(B\) are on the shoreline of Lake George. A surveyor is located at a third point \(C\) some distance from both points. The distance from \(A\) to \(C\) is 180.0 meters and the distance from \(B\) to \(C\) is 120.0 meters. The surveyor determines that the measure of \(\angle A C B\) is \(56.3^{\circ} .\) To the nearest tenth of a meter, what is the distance from \(A\) to \(B ?\)

For each \(\triangle O R S, O\) is the origin, \(R\) is on the positive ray of the \(x\) -axis and \(\overline{P S}\) is the altitude from \(S\) to \(\overrightarrow{O R}\) . Find the exact coordinates of \(R\) and \(S .\) b. Find the exact area of \(\triangle O R S .\) \(O R=5, \mathrm{m} \angle R O S=\frac{\pi}{3}, O S=3\)

From a point 50 feet from the foot of a vertical monument, the measure of the angle of elevation of the top of the monument is 65 degrees. What is the height of the monument to the nearest foot?

Two sailboats leave a dock at the same time sailing on courses that form an angle of \(112^{\circ}\) with each other. If one boat sails at 10.0 knots per hour and the other sails at 12.0 knots per hour, how many nautical miles apart are the boats after two hours? (nautical miles \(=\) knots \(\times\) time) Round to the nearest tenth.

A vertical telephone pole that is 15 feet high is braced by two wires from the top of the pole to two points on the ground that are 5.0 feet apart on the same side of the pole and in a straight line with the foot of the pole. The shorter wire makes an angle of 65 degrees with the ground. Find the length of each wire to the nearest tenth.

Use the Law of Cosines to prove that if the angle between two congruent sides of a triangle measures \(60^{\circ},\) the triangle is equilateral.

For each \(\triangle O R S, O\) is the origin, \(R\) is on the positive ray of the \(x\) -axis and \(\overline{P S}\) is the altitude from \(S\) to \(\overrightarrow{O R}\) . Find the exact coordinates of \(R\) and \(S .\) b. Find the exact area of \(\triangle O R S .\) \(O R=8, \mathrm{m} \angle R O S=\frac{3 \pi}{4}, O S=8\)

From point \(C\) at the top of a cliff, two points, \(A\) and \(B,\) are sited on level ground. Points \(A\) and \(B\) are on a straight line with \(D,\) a point directly below \(C .\) The angle of depression of the nearer point, \(A,\) is 72 degrees and the angle of depression of the farther point, \(B,\) is 48 degree. If the points \(A\) and \(B\) are 20 feet apart, what is the height of the cliff to the nearest foot?

For each \(\triangle O R S, O\) is the origin, \(R\) is on the positive ray of the \(x\) -axis and \(\overline{P S}\) is the altitude from \(S\) to \(\overrightarrow{O R}\) . Find the exact coordinates of \(R\) and \(S .\) b. Find the exact area of \(\triangle O R S .\) \(O R=20, O S=R S, P S=10\)

Mark is building a kite that is a quadrilateral with two pairs of congruent adjacent sides. One diagonal divides the kite into two unequal isosceles triangles and measures 14 inches. Each leg of one of the isosceles triangles measures 5 inches and each leg of the other measures 12 inches. Find the measures of the four angles of the quadrilateral.

For each \(\triangle O R S, O\) is the origin, \(R\) is on the positive ray of the \(x\) -axis and \(\overline{P S}\) is the altitude from \(S\) to \(\overrightarrow{O R}\) . Find the exact coordinates of \(R\) and \(S .\) b. Find the exact area of \(\triangle O R S .\) \(O R=7, \mathrm{m} \angle R O S=\frac{\pi}{6}, P S=8\)