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

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Question

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?

Answer

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

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Question

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

Answer

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

_x_2 + 62 = 102

Now we solve for x:

_x_2 + 36 = 100

_x_2 = 100 – 36

_x_2 = 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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Question

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?

Answer

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

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Question

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

Answer

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

_x_2 + 22 = (√8)2 = 8

Now we solve for x:

_x_2 + 4 = 8

_x_2 = 8 – 4

_x_2 = 4

x = 2

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Question

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 $\overline{BD}$ .

Answer

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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Question

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?

Answer

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:

State the theorem.

Substitute your variables.

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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