Right Triangle Relationships and Trigonometry Unit Test 100%

Given right triangle RST, what is the value of sin(S)?

5/13

Given right triangle MNO, which represents the value of cos(M)?

MN/MO

Which equation can be used to find the length of AC?

(10)sin(40o) = AC

Which set of numbers can represent the side lengths, in centimeters, of a right triangle?

10, 24, 26

A ramp leading into a building makes a 15° angle with the ground. The end of the ramp is 10 feet from the base of the building. Approximately how long is the ramp? Round to the nearest tenth.

10.4 feet

The equation tan(55°) = 15/b can be used to find the length of AC. What is the length of AC? Round to the nearest tenth.

10.5 in.

What is the length of AC? Round to the nearest tenth.

10.5 m

What is the value of n to the nearest whole number?

22

Building A and building B are 500 meters apart. There is no road between them, so to drive from building A to building B, it is necessary to first drive to building C and then to building B. About how much farther is it to drive than to walk directly from building A to building B? Round to the nearest whole number.

183 meters

To support a tree damaged in a storm, a 12-foot wire is secured from the ground to the tree at a point 10 feet off the ground. The tree meets the ground at a right angle. At approximately what angle does the wire meet the ground?

56.4°

Which is the best approximation for the measure of angle ABC?

58.3°

What is the measure of ∠C to the nearest whole degree?

77°

Given right triangle XYZ, which correctly describes the locations of the sides in relation to ∠Y?

a is adjacent, b is opposite, c is the hypotenuse

Given right triangle JKM, which correctly describes the locations of the sides in relation to ∠J?

a is the hypotenuse, b is adjacent, c is opposite

Which equation is correct and can be used to solve for the value of z?

sin(51°)/2.6 = sin(76°)/z

Given right triangle XYZ, what is the value of tan(Y)?

√3/3

What is the approximate value of y − x?

17.2°

The equation cos(35°)=a/25 can be used to find the length of BC. What is the length of BC? Round to the nearest tenth.

20.5 in.

What is the approximate measure of angle K? Use the law of sines to find the answer.

34°

In right triangle XYZ, the right angle is located at vertex Y. The length of line segment XY is 12.4 cm. The length of line segment YZ is 15.1 cm. Which is the approximate measure of angle YZX?

39.4°

What is the length of the side opposite ∠B?

4 units

Use the law of sines to find the value of w. What is the best approximation of the value of w?

4.0 cm

What is the approximate value of x? Round to the nearest tenth.

4.6 cm

Given right triangle MNL, what is the value of cos(M)?

4/5

Which is the best approximation for the measure of angle EGF?

40.2°

Given right triangle DEF, what is the value of tan(F)?

40/9

What is the approximate measure of angle F? Use the law of sines to find the answer.

44.4°

Which equation correctly applies the law of cosines to solve for an unknown angle measure?

82 = 72 + 112 - 2(7)(11)cos(P)

On which triangle can the law of cosines be used to find the length of an unknown side?

B

In which triangle is the value of x equal to tan−1(3.1/5.2)?

D

On which triangle can the law of cosines be applied once to find an unknown angle measure?

D

The length of segment EF is 12 cm. Which statements regarding triangle DEF are correct? Select three options.

EF is the longest side of △DEF. DF = 6 cm DE = 6√3 cm

Which equation can be used to find the measure of angle LJK?

cos(x) = 10/15

Consider the diagram and the proof below. Given: In △ABC, AD ⊥ BC Prove: sin(B)/b = sin(C)/c What is the missing statement in Step 6?

csin(B) = bsin(C)

Abby used the law of cosines for △KMN to solve for k. k2 = 312 + 532 - 2(31)(53)cos(37°) What additional information did Abby know that is not shown in the diagram?

m∠K = 37° and n = 31

Which classification best represents a triangle with side lengths 6 cm, 10 cm, and 12 cm?

obtuse, because 62 + 102 < 122

Which equation correctly uses the law of cosines to solve for the missing side length of △PQR?

p2 = 62 + 82 - 2(6)(8)cos(39°)

Which equation is true for triangle QRS?

sin(100°)/3.5 = sin(s)=2.4

Right triangle ABC is shown. Which equation can be used to solve for c?

sin(50°) = 3/c

Consider the proof. Given: In △ABC, BD ⊥ AC Prove: the formula for the law of cosines, a2 = b2 + c2 - 2bccos(A) What is the missing reason in Step 8?

substitution

An acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation of the possible range of values for the third side, s?

6 < s < 12.8

The longest side of an obtuse triangle measures 20 cm. The two shorter sides measure x cm and 3x cm. Rounded to the nearest tenth, what is the greatest possible value of x?

6.3

Which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.?

acute, because 102+122>152

Julian describes an angle in the triangle using these statements. GH is the adjacent side. HK is the opposite side. GK is the hypotenuse. Which angle(s) is Julian describing?

∠G

Using the side lengths of △PQR and △STU, which angle has a sine ratio of 4/5?

∠P

What is the length of line segment RS? Use the law of sines to find the answer. Round to the nearest tenth.

2.4 units

A triangle has angles that measure 30°, 60°, and 90°. The hypotenuse of the triangle measures 10 inches. Which is the best estimate for the perimeter of the triangle? Round to the nearest tenth.

23.7 in.

Use the law of sines to find the value of a. What is the best approximation of the value of a?

3.0 cm

A support beam needs to be placed at a 28° angle of elevation so that the top meets a vertical beam 1.6 meters above the horizontal floor. The vertical beam meets the floor at a 90° angle. Approximately how far from the vertical beam should the lower end of the support beam be placed along the horizontal floor?

3.0 meters

Jules owns a square plot of land that measures 30 yards on each side. He plans to divide the land in half by building a fence, as shown by the dotted line below. How many yards of fencing will Jules need?

30√2 yd

What is the length of AB? Round to the nearest tenth.

38.6 m

A parallelogram has side lengths of 13 and 17 and an angle that measures 64°. What is x, the length of the diagonal, to the nearest whole number?

16

Use the law of sines to find the value of y. Round to the nearest tenth.

2.5 units

Two sides of an acute triangle measure 5 inches and 8 inches. The length of the longest side is unknown. What is the greatest possible whole-number length of the unknown side?

9 inches

What is the approximate perimeter of the triangle? Use the law of sines to find the answer.

9.2 units

In which triangle is the value of x equal to cos−1(4.3/6.7)?

A

Consider triangle QRS. The legs each have a length of 10 units. What is the length of the hypotenuse of the triangle?

10√2 units

The equation sin(25°) = 9/c can be used to find the length of AB. What is the length of AB? Round to the nearest tenth.

21.3 in.

Triangle MNO is an equilateral triangle with sides measuring 16√3 units. What is the height of the triangle?

24 units

An acute triangle has side lengths 21 cm, x cm, and 2x cm. If 21 is one of the shorter sides of the triangle, what is the greatest possible length of the longest side, rounded to the nearest tenth?

24.2 cm

The hypotenuse of a 45°-45°-90° triangle measures 7√2 units. What is the length of one leg of the triangle?

7 units

Which statements are true about triangle QRS? Check all that apply.

The side opposite ∠Q is RS The hypotenuse is QR. The side adjacent to ∠Q is QS.

Two students describe the sides of right triangle ABC in relation to ∠B. Tomas AB is the hypotenuse. AC is the opposite side. BC is the adjacent side. Iliana AB is the hypotenuse. BC is the opposite side. AC is the adjacent side. Who is correct? Explain.

Tomas is correct; AC is opposite ∠B and BC is adjacent to ∠B.

Which relationship in the triangle must be true?

sin(B) = cos(90 - B)

Which trigonometric ratios are correct for triangle ABC? Check all that apply.

sin(C) = √3/2 tan(C) = √3 sin(B) = 1/2

Which equation can be used to find the measure of angle FGE?

sin(x) = 9/15.2

Triangle ABC is a right triangle and cos(22.6°)= b/13. Solve for b and round to the nearest whole number. Which equation correctly uses the value of b to solve for a?

tan(22.6°) = a/12

Given △DEF, which is not equal to cos(F)?

tan(F).

The equation sin(40°)=b/20 can be used to determine the length of line segment AC. What is the length of AC? Round to the nearest tenth.

12.9 cm

Given △ABC ~ △XYZ, what is the value of cos(Z)?

12/13

Given right triangle DEF, what is the value of sin(E)?

3/5

A wall in Maria's bedroom is in the shape of a trapezoid. The wall can be divided into a rectangle and a triangle. Using the 45°-45°-90° triangle theorem, find the value of h, the height of the wall.

13 ft

Kari is flying a kite. She releases 50 feet of string. What is the approximate difference in the height of the kite when the string makes a 25° angle with the ground and when the string makes a 45° angle with the ground? Round to the nearest tenth.

14.2 feet

What is the length of BC? Round to the nearest tenth.

14.5 cm

Consider triangle PQR. What is the length of side QR?

16 units

Which equation can be used to solve for the measure of angle ABC?

tan(x) = 2.4/10

Which equation can be used to find the measure of angle BAC?

tan−1(12/5) = x

Given right triangle ABC, what is the value of tan(A)?

12/5

Line segment YV of rectangle YVWX measures 24 units. What is the length of line segment YX?

8√3 units

Use the diagram to complete the statement. Given △JKL, sin(38°) equals

cos(52°).

Which equation can be solved to find one of the missing side lengths in the triangle?

cos(60°) = a/12

Which equation can be used to find the measure of angle GFE?

cos−1(11.9/14.5) = 0

What are the angle measures of triangle ABC?

m∠A = 90°, m∠B = 60°, m∠C = 30°

A right triangle has side lengths AC = 7 inches, BC = 24 inches, and AB = 25 inches. What are the measures of the angles in triangle ABC?

m∠A ≈ 73.7°, m∠B ≈ 16.3°, m∠C ≈ 90°

What is the measure of ∠S to the nearest whole degree?

33°

Each leg of a 45°-45°-90° triangle measures 12 cm. What is the length of the hypotenuse?

12√2 cm

A right triangle has a 30° angle. The leg adjacent to the 30° angle measures 25 inches. What is the length of the other leg? Round to the nearest tenth.

14.4 in.

A right triangle has one angle that measure 23°. The adjacent leg measures 27.6 cm and the hypotenuse measures 30 cm. What is the approximate area of the triangle? Round to the nearest tenth.

161.7 cm2

Two sides of an obtuse triangle measure 10 inches and 15 inches. The length of longest side is unknown. What is the smallest possible whole-number length of the unknown side?

19 inches

The length of the hypotenuse, line segment GH, in triangle GJH measures 6 cm. Line segment JH measures 2 cm. Which is the approximate measure of angle JGH?

19.5°

The measure of angle BAC can be calculated using the equation sin-1(3.1/4.5)=x. What is the measure of angle BAC? Round to the nearest whole degree.

44°

The equation tan−1(8.9/7.7) = x can be used to find the measure of angle LKJ. What is the measure of angle LKJ? Round to the nearest whole degree.

49

Which set of numbers can represent the side lengths, in inches, of an acute triangle?

5, 7, 8

The longest side of an acute isosceles triangle is 8 centimeters. Rounded to the nearest tenth, what is the smallest possible length of one of the two congruent sides?

5.7 cm

Given right triangle JKL, what is the value of cos(L)?

5/13

The law of cosines is used to find the measure of ∠Z. To the nearest whole degree, what is the measure of ∠Z?

51º

The length of the hypotenuse, line segment AC, in right triangle ABC is 25 cm. The length of line segment BC is 15 cm. Which is the approximate measure of angle ACB?

53.1°

Which is the approximate measure of angle Y? Use the law of sines to find the answer.

59°

Consider triangle GHJ. What is the length of line segment HJ?

5√3 units

The shorter sides of an acute triangle are x cm and 2x cm. The longest side of the triangle is 15 cm. What is the smallest possible whole-number value of x?

7

Two sides of an obtuse triangle measure 12 inches and 14 inches. The longest side measures 14 inches. What is the greatest possible whole-number length of the unknown side?

7 inches

A beach has two floating docks. One is 650 meters east of the lifeguard stand. The other is 60° southeast and 750 meters from the lifeguard stand. Rounded to the nearest meter, what is the distance between the docks? Round to the nearest meter.

705 meters

The longest side of an acute triangle measures 30 inches. The two remaining sides are congruent, but their length is unknown. What is the smallest possible perimeter of the triangle, rounded to the nearest tenth?

72.4 in.

Which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle?

8, 10, 14

The length of segment XY is 9 cm. Which statements regarding triangle XYZ are correct? Select two options.

YZ = 9 cm XZ = 9√2 cm

Joey is building a frame for a sandbox. The sandbox is going to be a quadrilateral that has the lengths shown. If the diagonal of the sandbox measures 14 feet, which best describes the shape of the sandbox?

a quadrilateral, because angle C and angle X are acute