GMAT Math : Calculating the length of the hypotenuse of a right triangle : Pythagorean Theorem

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Example Question #12 : Right Triangles

What is the length of a right triangle hypotenuse if the the other two sides are and units long?

Possible Answers:

Correct answer:

Explanation:

To find a length of a side in a right triangle, we can use the Pythagorean theorem:

a^2 + b^2 = c^2

5^2 + 12^2 = c^2

25 + 144 = c^2

169 = c^2

13 = c

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Example Question #1 : Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

What is the hypotenuse of a right triangle with sides 14 and 48?

Possible Answers:

$\dpi{100} \small 55$

$\dpi{100} \small 25$

$\dpi{100} \small 48$

$\dpi{100} \small 52$

$\dpi{100} \small 50$

Correct answer:

$\dpi{100} \small 50$

Explanation:

For starters, we know that the hypotenuse is the longest side of a triangle, so we can immediately cross off 25 as an answer choice because it is smaller than one of the legs, 48. To find the hypotenuse, we can use the Pythagorean Theorem.

$hypotenuse^{^{2}} = 14^{2} + 48^{2} = 2500$

$hypotenuse = 50$

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Example Question #14 : Right Triangles

A right triangle has a height of 8 and a base of 15. What is the length of its hypotenuse?

Possible Answers:

Correct answer:

Explanation:

We are given the length of the height and the base, so we can use the Pythagorean theorem to calculate the length of the hypotenuse:

a^2+b^2=c^2

8^2+15^2=c^2

c^2=64+225

$c^2=289\rightarrow c=\sqrt{289}=17$

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Example Question #12 : Right Triangles

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

Possible Answers:

$\sqrt{21}$

$\sqrt{74}$

$\sqrt{30}$

$\sqrt{15}$

$\sqrt{75}$

Correct answer:

$\sqrt{74}$

Explanation:

In order to find the value of the hypotenuse , we need to plug the values of the legs and into the Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$

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

$25+49=c^{2}$

$74=c^{2}$

$\sqrt{74}=c$

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Example Question #16 : Triangles

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

Possible Answers:

Correct answer:

Explanation:

In order to find the value of the hypotenuse given area and leg length , we first need to calculate the length of the other leg length . Given the area equation $A=\frac{1}{2}ab$ :

2A=ab

$\frac{2A}{b}=a$

Plugging in and :

$\frac{2(30)}{12}=a$

$\frac{60}{12}=a$

5=a

Now we need to plug the values of the legs and into the Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$

$5^{2}+12^{2}=c^{2}$

$25+144=c^{2}$

$169=c^{2}$

13=c

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Example Question #4 : Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

What is the hypotenuse of a right triangle with legs of lengths 7cm and 9cm ?

Possible Answers:

$\sqrt{65}cm$

15cm

$\sqrt{130}cm$

$\sqrt{70}cm$

12cm

Correct answer:

$\sqrt{130}cm$

Explanation:

In order to find the value of the hypotenuse , we need to plug the values of the legs and into the Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$

$(7cm)^{2}+(9cm)^{2}=c^{2}$

$49cm+81cm=c^{2}$

$130cm=c^{2}$

$\sqrt{130}cm=c$

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Example Question #3 : Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Each leg of a $\textup{45-45-90}$ triangle has length $2 ^{K}$ . Give the length of its hypotenuse.

Possible Answers:

$2 ^{K +1}$

$2 ^{ 2K }$

$2 ^{ \frac{1}{2}K }$

$2 ^{K + \frac{1}{2}}$

$2 ^{K - \frac{1}{2}}$

Correct answer:

$2 ^{K + \frac{1}{2}}$

Explanation:

The hypotenuse of a $\textup{45-45-90}$ triangle has length $\sqrt{2} = 2^{\frac{1}{2}}$ times that of a leg, which here is

$2 ^{K} \cdot 2^{\frac{1}{2}} = 2 ^{K+ \frac{1}{2}}$ .

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GMAT Math : Calculating the length of the hypotenuse of a right triangle : Pythagorean Theorem

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Example Questions

Example Question #12 : Right Triangles

Example Question #1 : Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #14 : Right Triangles

Example Question #12 : Right Triangles

Example Question #16 : Triangles

Example Question #4 : Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #3 : Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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