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SSAT Upper Level Math : Properties of Triangles

Study concepts, example questions & explanations for SSAT Upper Level Math

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All SSAT Upper Level Math Resources

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Example Questions

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

A right triangle has a hypotenuse of $\displaystyle 12$ and one leg has a length of $\displaystyle 9$ . What is the length of the other leg?

Possible Answers:

$\displaystyle 11$

$\displaystyle 8$

$7\sqrt{3}$ $\displaystyle 7\sqrt{3}$

$\displaystyle 10$

$3\sqrt{7}$ $\displaystyle 3\sqrt{7}$

Correct answer:

$3\sqrt{7}$ $\displaystyle 3\sqrt{7}$

Explanation:

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

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

Plugging in our given values:

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

Subtracting 9^2 $\displaystyle 9^2$ from each side of the equation:

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

$b^{2}=144-81$ $\displaystyle b^{2}=144-81$

$b^{2}=63$ $\displaystyle b^{2}=63$

Taking the square root of each side of the equation:

$b=\sqrt{63}$ $\displaystyle b=\sqrt{63}$

Simplifying the square root:

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

$b=3\sqrt{7}$ $\displaystyle b=3\sqrt{7}$

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

A right triangle has two legs of length $\displaystyle 6$ and $\displaystyle 10$ , respectively. What is the hypotenuse of the right triangle?

Possible Answers:

$\displaystyle 12$

$2\sqrt{17}$ $\displaystyle 2\sqrt{17}$

$2\sqrt{34}$ $\displaystyle 2\sqrt{34}$

$17\sqrt{2}$ $\displaystyle 17\sqrt{2}$

$34\sqrt{2}$ $\displaystyle 34\sqrt{2}$

Correct answer:

$2\sqrt{34}$ $\displaystyle 2\sqrt{34}$

Explanation:

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

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

Plugging in our given values:

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

$36+100=c^{2}$ $\displaystyle 36+100=c^{2}$

$136=c^{2}$ $\displaystyle 136=c^{2}$

$\sqrt{136}=c$ $\displaystyle \sqrt{136}=c$

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

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

$2\sqrt{34}=c$ $\displaystyle 2\sqrt{34}=c$

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

A right triangle has a leg of length $\displaystyle 7$ and a hypotenuse of length $\displaystyle 11$ . What is the length of the other leg?

Possible Answers:

$2\sqrt{6}$ $\displaystyle 2\sqrt{6}$

$6\sqrt{2}$ $\displaystyle 6\sqrt{2}$

$9\sqrt{6}$ $\displaystyle 9\sqrt{6}$

$6\sqrt{9}$ $\displaystyle 6\sqrt{9}$

$\displaystyle 9$

Correct answer:

$6\sqrt{2}$ $\displaystyle 6\sqrt{2}$

Explanation:

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

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

Plugging in our given values:

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

Subtracting 7^2 $\displaystyle 7^2$ from each side of the equation:

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

$b^{2}=121-49$ $\displaystyle b^{2}=121-49$

$b^{2}=72$ $\displaystyle b^{2}=72$

$b=\sqrt{72}$ $\displaystyle b=\sqrt{72}$

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

$b=6\sqrt{2}$ $\displaystyle b=6\sqrt{2}$

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

Find the length of the missing side.

Possible Answers:

160 $\displaystyle 160$

$4\sqrt{10}$ $\displaystyle 4\sqrt{10}$

$\sqrt{15}$ $\displaystyle \sqrt{15}$

$\displaystyle 80$

Correct answer:

$4\sqrt{10}$ $\displaystyle 4\sqrt{10}$

Explanation:

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

$\displaystyle x^2+6^2=14^2$

$\displaystyle x^2+36=196$

x^2=160 $\displaystyle x^2=160$

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

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the missing side.

Possible Answers:

$8\sqrt2$ $\displaystyle 8\sqrt2$

176 $\displaystyle 176$

$4\sqrt{13}$ $\displaystyle 4\sqrt{13}$

$4\sqrt{11}$ $\displaystyle 4\sqrt{11}$

Correct answer:

$4\sqrt{11}$ $\displaystyle 4\sqrt{11}$

Explanation:

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

$\displaystyle x^2+7^2=15^2$

$\displaystyle x^2+49=225$

x^2=176 $\displaystyle x^2=176$

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

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

Find the length of the missing side.

Possible Answers:

$\displaystyle 25$

$\displaystyle 4$

$\displaystyle 8$

$\displaystyle 5$

Correct answer:

$\displaystyle 5$

Explanation:

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

$\displaystyle x^2+12^2=13^2$

$\displaystyle x^2+144=169$

x^2=25 $\displaystyle x^2=25$

$x=\sqrt{25}=5$ $\displaystyle x=\sqrt{25}=5$

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Example Question #4 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the missing side.

Possible Answers:

$\displaystyle 14$

100 $\displaystyle 100$

$\displaystyle 12$

$\displaystyle 10$

Correct answer:

$\displaystyle 10$

Explanation:

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

$\displaystyle x^2+24^2=26^2$

$\displaystyle x^2+576=676$

x^2=100 $\displaystyle x^2=100$

$x=\sqrt{100}=10$ $\displaystyle x=\sqrt{100}=10$

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Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the missing side.

Possible Answers:

$\displaystyle 64$

$\displaystyle 8$

$\displaystyle 9$

$\displaystyle 5$

Correct answer:

$\displaystyle 8$

Explanation:

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

$\displaystyle x^2+6^2=10^2$

$\displaystyle x^2+36=100$

x^2=64 $\displaystyle x^2=64$

$x=\sqrt{64}=8$ $\displaystyle x=\sqrt{64}=8$

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Example Question #6 : How To Find The Length Of The Side Of A Right Triangle

Find the length of the missing side.

Possible Answers:

$\displaystyle 20$

$10\sqrt3$ $\displaystyle 10\sqrt3$

300 $\displaystyle 300$

$10\sqrt5$ $\displaystyle 10\sqrt5$

Correct answer:

$10\sqrt3$ $\displaystyle 10\sqrt3$

Explanation:

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

$\displaystyle x^2+10^2=20^2$

$\displaystyle x^2+100=400$

x^2=300 $\displaystyle x^2=300$

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

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Example Question #11 : Properties Of Triangles

Find the length of the missing side.

Possible Answers:

400 $\displaystyle 400$

$\displaystyle 20$

$\displaystyle 12$

$\displaystyle 15$

Correct answer:

$\displaystyle 20$

Explanation:

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

$\displaystyle x^2+15^2=25^2$

$\displaystyle x^2+225=625$

x^2=400 $\displaystyle x^2=400$

$x=\sqrt{400}=20$ $\displaystyle x=\sqrt{400}=20$

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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : Properties of Triangles

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

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

Example Question #2 : 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

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

Example Question #2 : 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

Example Question #4 : How To Find The Length Of The Side Of A Right Triangle

Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

Example Question #6 : How To Find The Length Of The Side Of A Right Triangle

Example Question #11 : Properties Of Triangles

All SSAT Upper Level Math Resources

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