How to find the length of the side of a right triangle - SSAT Upper Level Quantitative

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Question

A right triangle has a hypotenuse of 39 and one leg is 36. What is the length of the other leg?

Answer

You may recognize these numbers as multiples of 13 and 12 (each by a factor of 3) and remember that sides of length 5, 12 and 13 make a special right triangle. So the other leg would be 15 $(5\times 3)$ .

If you don't remember this, you can use Pythagorean theorem:

$39^{2}-36^{2}= 1521-1296=225$

$\sqrt{225}=15$

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Question

A right triangle has a hypotenuse of and one leg has a length of . What is the length of the other leg?

Answer

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

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

Plugging in our given values:

$9^{2}+b^{2}=12^{2}$

Subtracting from each side of the equation:

$b^{2}=12^{2}-9^{2}$

$b^{2}=144-81$

$b^{2}=63$

Taking the square root of each side of the equation:

$b=\sqrt{63}$

Simplifying the square root:

$b=\sqrt{9\times 7}$

$b=3\sqrt{7}$

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Question

A right triangle has two legs of length and , respectively. What is the hypotenuse of the right triangle?

Answer

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

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

Plugging in our given values:

$6^{2}+10^{2}=c^{2}$

$36+100=c^{2}$

$136=c^{2}$

$\sqrt{136}=c$

$\sqrt{17\times 8}=c$

$\sqrt{17\times 4\times 2}=c$

$2\sqrt{34}=c$

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Question

A right triangle has a leg of length and a hypotenuse of length . What is the length of the other leg?

Answer

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

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

Plugging in our given values:

$7^{2}+b^{2}=11^{2}$

Subtracting from each side of the equation:

$b^{2}=11^{2}-7^{2}$

$b^{2}=121-49$

$b^{2}=72$

$b=\sqrt{72}$

$b=\sqrt{36\times 2}$

$b=6\sqrt{2}$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{160}=4\sqrt{10}$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{176}=4\sqrt{11}$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{25}=5$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{100}=10$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{64}=8$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{300}=10\sqrt{3}$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{400}=20$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{1296}=36$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{225}=15$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{900}=30$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{196}=14$

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Question

Find the length of the missing side.

Answer

Use the Pythagorean Theorem to find the length of the missing side.

$x=\sqrt{3600}=60$

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Question

A right triangle has a hypotenuse of 39 and one leg is 36. What is the length of the other leg?

Answer

You may recognize these numbers as multiples of 13 and 12 (each by a factor of 3) and remember that sides of length 5, 12 and 13 make a special right triangle. So the other leg would be 15 $(5\times 3)$ .

If you don't remember this, you can use Pythagorean theorem:

$39^{2}-36^{2}= 1521-1296=225$

$\sqrt{225}=15$

Compare your answer with the correct one above

Question

A right triangle has a hypotenuse of and one leg has a length of . What is the length of the other leg?

Answer

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

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

Plugging in our given values:

$9^{2}+b^{2}=12^{2}$

Subtracting from each side of the equation:

$b^{2}=12^{2}-9^{2}$

$b^{2}=144-81$

$b^{2}=63$

Taking the square root of each side of the equation:

$b=\sqrt{63}$

Simplifying the square root:

$b=\sqrt{9\times 7}$

$b=3\sqrt{7}$

Compare your answer with the correct one above

Question

A right triangle has two legs of length and , respectively. What is the hypotenuse of the right triangle?

Answer

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

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

Plugging in our given values:

$6^{2}+10^{2}=c^{2}$

$36+100=c^{2}$

$136=c^{2}$

$\sqrt{136}=c$

$\sqrt{17\times 8}=c$

$\sqrt{17\times 4\times 2}=c$

$2\sqrt{34}=c$

Compare your answer with the correct one above

Question

A right triangle has a leg of length and a hypotenuse of length . What is the length of the other leg?

Answer

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

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

Plugging in our given values:

$7^{2}+b^{2}=11^{2}$

Subtracting from each side of the equation:

$b^{2}=11^{2}-7^{2}$

$b^{2}=121-49$

$b^{2}=72$

$b=\sqrt{72}$

$b=\sqrt{36\times 2}$

$b=6\sqrt{2}$

Compare your answer with the correct one above

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