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

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Given a right triangle with a leg length of 6 and a hypotenuse length of 10, find the length of the other leg, x.

Act_math_106

Possible Answers:

Correct answer:

Explanation:

Using Pythagorean Theorem, we can solve for the length of leg x:

x² + 6² = 10²

Now we solve for x:

x² + 36 = 100

x² = 100 – 36

x² = 64

x = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

In a right triangle a hypotenuse has a length of 8 and leg has a length of 7. What is the length of the third side to the nearest tenth?

Possible Answers:

3.6

1.0

3.9

2.4

Correct answer: 3.9

Explanation:

Using the pythagorean theorem, 8²=7²+x². Solving for x yields the square root of 15, which is 3.9

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Given a right triangle with a leg length of 2 and a hypotenuse length of √8, find the length of the other leg, x.

Vt_triangle_x-2-sqrt8

Possible Answers:

√8

Correct answer:

Explanation:

Using Pythagorean Theorem, we can solve for the length of leg x:

x² + 2² = (√8)²= 8

Now we solve for x:

x² + 4 = 8

x² = 8 – 4

x² = 4

x = 2

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

The legs of a right triangle are $8\ cm$ and $11\ cm$ . Rounded to the nearest whole number, what is the length of the hypotenuse?

Possible Answers:

$2\ cm$

$14\ cm$

$9\ cm$

$15\ cm$

$10\ cm$

Correct answer:

$14\ cm$

Explanation:

Use the Pythagorean Theorem. The sum of both legs squared equals the hypotenuse squared.

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Points $\dpi{100} \small A$ , $\dpi{100} \small B$ , and $\dpi{100} \small C$ are collinear (they lie along the same line). $\angle ACD = 90^{\circ}$ , $\angle CAD=30^{\circ}$ , $\angle CBD=60^{\circ}$ , $\overline{AD}=4$

Find the length of segment $\overline{BD}$ .

Possible Answers:

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

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

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

$2$

$2\sqrt{3}$

Correct answer:

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

Explanation:

The length of segment $\overline{BD}$ is $\frac{4\sqrt{3}}{3}$

Note that triangles $\dpi{100} \small ACD$ and $\dpi{100} \small BCD$ are both special, 30-60-90 right triangles. Looking specifically at triangle $\dpi{100} \small ACD$ , because we know that segment $\overline{AD}$ has a length of 4, we can determine that the length of segment $\overline{CD}$ is 2 using what we know about special right triangles. Then, looking at triangle $\dpi{100} \small BCD$ now, we can use the same rules to determine that segment $\overline{BD}$ has a length of $\frac{4}{\sqrt{3}}$

which simplifies to $\frac{4\sqrt{3}}{3}$ .

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Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

A handicap ramp is $32\textup{ feet}$ long, and a person traveling the length of the ramp goes up $4\textup{ feet}$ vertically. What horizontal distance does the ramp cover?

Possible Answers:

$13\sqrt4\textup{ feet}$

$12\sqrt7\textup{ feet}$

$13\sqrt9\textup{ feet}$

$27\textup{ feet}$

$5\sqrt13\textup{ feet}$

Correct answer:

$12\sqrt7\textup{ feet}$

Explanation:

In this case, we are already given the length of the hypotenuse of the right triangle, but the Pythagorean formula still helps us. Plug and play, remembering that must always be the hypotenuse:

a^2 + b^2 = c^2 State the theorem.

Substitute your variables.

a^2 = 1008 Simplify.

$a = \sqrt{1008} = \sqrt{144\cdot 7} = 12\sqrt7$

Thus, the ramp covers $12\sqrt7\textup{ feet}$ of horizontal distance.

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

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

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