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

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

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

CORRECT

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:

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

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

CORRECT

Explanation

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

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

$\sqrt{130}cm$

CORRECT

$\sqrt{65}cm$

$\sqrt{70}cm$

Explanation

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

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

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

$130cm=c^{2}$

$\sqrt{130}cm=c$

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

$\sqrt{74}$

CORRECT

$\sqrt{75}$

$\sqrt{15}$

$\sqrt{21}$

$\sqrt{30}$

Explanation

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

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

$25+49=c^{2}$

$74=c^{2}$

$\sqrt{74}=c$

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

CORRECT

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 $A=\frac{1}{2}ab$ :

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

Plugging in and :

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

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

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

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

$25+144=c^{2}$

$169=c^{2}$

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

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

CORRECT

$2 ^{K +1}$

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

$2 ^{ 2K }$

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

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

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

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

$\dpi{100} \small 50$

CORRECT

$\dpi{100} \small 48$

$\dpi{100} \small 25$

$\dpi{100} \small 52$

$\dpi{100} \small 55$

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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