Bluebook SAT Test Right Triangles and Trigonometry 12

Question 1

In right triangle MNP with right angle at N, if MN = 9 and tan(M) = 9/40, what is MP?

Accepted Answer

41

Answer

tan(M) = NP/MN = NP/9 = 9/40, so NP = (9 × 9)/40 = 81/40. Then MP = √(9² + (81/40)²). Actually, tan(M) = opposite/adjacent = NP/9 = 9/40, so NP = 81/40 ≈ 2.025. MP = √(81 + (81/40)²). Let me recalculate: if the ratio is 9:40 for legs, then legs could be 9 and 40, hypotenuse = √(81+1600) = √1681 = 41.

Question 2

A right triangle has vertices at coordinates (0,0), (6,0), and (0,8). What is the sine of the angle at the origin?

Accepted Answer

8/10

Answer

The legs are 6 and 8, so the hypotenuse = √(36+64) = 10. The angle at the origin has opposite side = 8 (vertical leg) and hypotenuse = 10, so sin = 8/10.

Question 3

In a right triangle, the hypotenuse is 20 and one acute angle is 30°. What is the area of the triangle?

Accepted Answer

50√3

Answer

The legs are 20·sin(30°) = 10 and 20·cos(30°) = 10√3. Area = (1/2)(10)(10√3) = 50√3.

Question 4

Two buildings stand 50 meters apart. From the top of the shorter building (height h), the angle of elevation to the top of the taller building is 45°. If the taller building is 80 meters tall, what is h?

Accepted Answer

30 m

Answer

tan(45°) = (80 − h)/50 = 1, so 80 − h = 50, giving h = 30 meters.

Question 5

In right triangle ABC with right angle at C, if the perimeter is 60 and the hypotenuse is 25, what is the area?

Accepted Answer

150

Answer

The sum of the two legs = 60 − 25 = 35. Let legs be a and b: a + b = 35 and a² + b² = 625. (a+b)² = a² + 2ab + b² = 1225, so 2ab = 1225 − 625 = 600, ab = 300. Area = (1/2)ab = 150.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Right Triangles and Trigonometry 12