SAT Math - How to find the length of the side of a right triangle

Solve each problem and decide which is the best of the choices given.

If $sin x =\frac{12}{13}$ , what is ?

$tan(x)=\frac{12}{5}$

$tan(x)=\frac{5}{13}$

$tan(x)=\frac{5}{12}$

$tan(x)=\frac{12}{13}$

$tan(x)=\frac{13}{12}$

Explanation

This is a triangle. We can find the value of the other leg by using the Pythagorean Theorem.

Remembering that

$sin=\frac{opposite}{hypotenuse}$ .

Thus,

If $sin(x)=\frac{12}{13}$ , you know the adjacent side is .

Thus, making

$tan(x)=\frac{opposite}{adjacent}=\frac{12}{5}$ because tangent is opposite/adjacent.

The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?

$7$

$5$

$6$

$9$

$11$

Explanation

$Area= \frac{1}{2}\times base\times height$

$42=\frac{1}{2}\times base\times 12$

$42=6\times base$

$base=7$

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

12 √6

33√2

Explanation

use the pythagorean theorem:

a2 + b2 = c2 ; a and b are sides, c is the hypotenuse

a2 + 1296 = 1521

a2 = 225

a = 15

A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?

Explanation

We can use the Pythagorean Theorem to solve for x.

92 + _x_2 = 152

81 + _x_2 = 225

_x_2 = 144

x = 12

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

Screen_shot_2013-03-18_at_10.29.39_pm

3.5

4.5

5.5

Explanation

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

13.07

14.87

171

Explanation

The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

Right triangle 7

Refer to the provided figure. Give the length of $\overline{BC}$ .

$10\sqrt{2}$

$10\sqrt{3}$

$\frac{20}{3} \sqrt{3}$

$\frac{20}{3} \sqrt{2}$

Explanation

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg $\overline{BC}$ is equal to that of hypotenuse $\overline{AC}$ , the length of which is 20. Therefore,

$BC = \frac{AC}{\sqrt{2}} = \frac{20}{\sqrt{2}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{2}$ :

$BC = \frac{20 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{20 \sqrt{2}}{2} = 10 \sqrt{2}$

A single-sided ladder is leaning against a wall. The angle between the end of the ladder that is on the ground and the ground itself is represented by $x^{\circ}$ . The ladder is sliding down the wall at a rate of 6 feet per second. If

$\sin x^{\circ}=\frac{3}{5}$

how many seconds does it take for the ladder to fall all the way to the ground? (The wall is a right angle to the ground.)

$0.5 \text{ seconds}$

$1 \text{ second}$

$6 \text{ seconds}$

$2.2 \text{ seconds}$

$1.1 \text{ seconds}$

Explanation

The ladder leaning against the wall forms a right triangle. The hypotenuse of the triangle is 5 ft., the length of the ladder.

Because sin x= opposite/hypotenuse, sine of the angle is equal to the length of the side opposite the angle divided by the hypotenuse. In this case, the length of the side opposite the angle is h, the height of the end of the ladder that is touching the wall. Thus,

$\sin x^{\circ} = \frac{h}{5}$

Because we are told that

$\sin x^{\circ}=\frac{3}{5}$

we know that h=3. Therefore, 3 feet is the height of the ladder. If the ladder is falling at a rate of 6 feet per second, we can find the number of seconds it will take the ladder to hit the ground with the equation

where h represents the height the ladder is falling from, and s represents the number of seconds it takes the ladder to fall. We can now solve for s:

$s=\frac{h}{6}$

$s=\frac{3}{6}=\frac{1}{2}=0.5$

It takes the ladder 0.5 seconds to fall to the ground.

Which of the following could NOT be the lengths of the sides of a right triangle?

12, 16, 20

8, 15, 17

5, 7, 10

5, 12, 13

14, 48, 50

Explanation

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2

Triangle