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

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Example Question #11 : How To Find The Length Of The Side Of A Right Triangle

Given $\bigtriangleup ABC$ with $m \angle A > m \angle B$ and $m \angle C = 90^{\circ }$ .

Which of the following could be the correct ordering of the lengths of the sides of the triangle?

I) AB > BC > AC

II) BC > AB > AC

III) BC > AC > AB

Possible Answers:

I or II only

II only

III only

I only

II or III only

Correct answer:

I only

Explanation:

Given two angles of unequal measure in a triangle, the side opposite the greater angle is longer than the side opposite the other angle. Since $m \angle A > m \angle B$ , it follows that BC > AC . Also, in a right triangle, the hypotenuse must be the longest side; in $\bigtriangleup ABC$ , since $\angle C$ is the right side, this hypotenuse is $\overline{AB}$ . It follows that AB > BC > AC , and that (I) is the only statement that can possibly be true.

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Example Question #11 : How To Find The Length Of The Side Of A Right Triangle

Triangle

$\left ( \bigtriangleup ABC \textup{ is a right triangle where } \angle B \textup{ is a right angle}\right)$

If $\angle A=30^{\circ}$ and $\overline{AB}=4$ , what is the length of $\overline{BC}$ ?

Possible Answers:

$\frac{4\sqrt{3}}{3}$

$\frac{4}{3}$

Correct answer:

$\frac{4\sqrt{3}}{3}$

Explanation:

AB is the leg adjacent to Angle A and BC is the leg opposite Angle A.

Since we have a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, the opposites sides of those angles will be in the ratio $x-x\sqrt{3}-2x$ .

Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to $x\sqrt{3}$ .

$4=x\sqrt{3}$

$\frac{4}{\sqrt{3}}=\frac{x\sqrt{3}}{\sqrt{3}}$

$\frac{4}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=x$

$x=\frac{4\sqrt{3}}{3}$

which also means

$\overline{BC}=\frac{4\sqrt{3}}{3}$

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Example Question #12 : How To Find The Length Of The Side Of A Right Triangle

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.)

Possible Answers:

$1 \text{ second}$

$6 \text{ seconds}$

$2.2 \text{ seconds}$

$0.5 \text{ seconds}$

$1.1 \text{ seconds}$

Correct answer:

$0.5 \text{ seconds}$

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

h=6s

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.

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Example Question #13 : How To Find The Length Of The Side Of A Right Triangle

Right triangle 7

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

Possible Answers:

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

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

$10\sqrt{2}$

$10\sqrt{3}$

Correct answer:

$10\sqrt{2}$

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 $\sqrt{2}$ :

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

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

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #11 : How To Find The Length Of The Side Of A Right Triangle

Example Question #11 : How To Find The Length Of The Side Of A Right Triangle

Example Question #12 : How To Find The Length Of The Side Of A Right Triangle

Example Question #13 : How To Find The Length Of The Side Of A Right Triangle

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