How to find the perimeter of a right triangle - SSAT Upper Level Quantitative (Math)

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Question

What is the perimeter of a right triangle with hypotenuse and a leg of length ?

Answer

Using the Pythagorean Theorem, the length of the second leg can be determined.

We are given the length of the hypotenuse and one leg.

$b^2=c^2-a^2\rightarrow b=\sqrt{c^2-a^2}$

$\sqrt{91^{2}-35^{2}} = \sqrt{8,281-1,225} = \sqrt{7,056} = 84$

The perimeter of the triangle is the sum of the lengths of the sides.

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Question

Which of these polygons has the same perimeter as a right triangle with legs 6 feet and 8 feet?

Answer

A right triangle with legs 6 feet and 8 feet has hypotentuse 10 feet, as this is a right triangle that confirms to the well-known Pythagorean triple 6-8-10. The perimeter is therefore feet, or 8 yards.

We are looking for a polygon with this perimeter. Each choice is a polygon with all sides one yard long, so we want the polygon with eight sides - the regular octagon is the correct choice.

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Question

The lengths of the legs of a right triangle are units and units. What is the perimeter of this right triangle?

Answer

First, we need to use the Pythagorean Theorem to find the hypotenuse of the triangle.

$14^2+48^2=\text{Hypotenuse}^2$

$196+2304=\text{Hypotenuse}^2$

$\text{Hypotenuse}^2=2500$

$\text{Hypotenuse}=\sqrt{2500}=50$

Now, add up all three side lengths to find the perimeter of the triangle.

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Question

A right triangle has leg lengths of and . Find the perimeter of this triangle.

Answer

First, use the Pythagorean Theorem to find the length of the hypotenuse.

Substituting in and for and (the lengths of the triangle's legs), we get:

$\sqrt{113}=\sqrt{c^2}$

$\sqrt{113}=c$

Now, add up the three sides to find the perimeter:

$7+8+\sqrt{113}=15+\sqrt{113}$

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Question

What is the perimeter of a right triangle with legs of length and , respectively?

Answer

In order to find the perimeter of the right triangle, we need to first find the missing length of the hypotenuse. In order to find the hypotenuse, use the Pythagorean Theorem:

$a^{2}+ b^{2}=c^{2}$ , where and are the lengths of the legs of the triangle, and is the length of the hypotenuse.

Substituting in our known values:

$5^{2}+ 7^{2}=c^{2}$

$25+ 49=c^{2}$

$74=c^{2}$

$\sqrt{74}=c$

Now that we have the lengths of all sides of the right triangle, we can calculate the perimeter:

$P=5+7+\sqrt{74}$

$P=12+\sqrt{74}$

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Question

What is the perimeter of a right triangle with a hypotenuse of length and a leg of length ?

Answer

In order to find the perimeter of the right triangle, we need to first find the missing length of the second leg. In order to find the second leg, use the Pythagorean Theorem:

$a^{2}+ b^{2}=c^{2}$ , where and are the lengths of the legs of the triangle, and is the length of the hypotenuse.

Substituting in our known values:

$8^{2}+ b^{2}=14^{2}$

Subtracting from each side of the equation lets us isolate the variable for which we are solving:

$b^{2}=14^{2}-8^{2}$

$b^{2}=196-64$

$b^{2}=132$

$b=\sqrt{132}$

$b=\sqrt{33\times 4}$

$b=2\sqrt{33}$

Now that we have the lengths of all three sides of the right triangle, we can calculate the perimeter :

$P=14+8+2\sqrt{33}$

$P=22+2\sqrt{33}$

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Question

Find the perimeter of a right triangle with two legs of length and , respectively.

Answer

In order to find the perimeter of the right triangle, we need to first find the missing length of the hypotenuse. In order to find the length of the hypotenuse, use the Pythagorean Theorem:

$a^{2}+ b^{2}=c^{2}$ , where and are the lengths of the legs of the triangle, and is the length of the hypotenuse.

Substituting in our known values:

$8^{2}+ 12^{2}=c^{2}$

$64+144=c^{2}$

$208=c^{2}$

$\sqrt{208}=c$

$\sqrt{13\times 16}=c$

$4\sqrt{13}=c$

Now that we have all three sides of the right triangle, we can calculate the perimeter:

$P=8+12+4\sqrt{13}$

$P=20+4\sqrt{13}$

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