GMAT Math : Calculating the length of the side of a right triangle

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Example Question #7 : Right Triangles

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

Possible Answers:

Correct answer:

Explanation:

We need to use the Pythagorean theorem:

a^2+b^2=c^2

a^2=c^2-b^2

a^2=10^2-8^2

a^2=100-64

a^2=36

a=6

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Example Question #1 : Calculating The Length Of The Side Of A Right Triangle

There is a big square that consists of four identical right triangles and a small square. If the area of the small square is 1, the area of the big square is 5, what is the length of the shortest side of the right triangles?

Possible Answers:

$\dpi{100} \small 3$

$\dpi{100} \small 4$

$\dpi{100} \small 5$

$\dpi{100} \small 2$

$\dpi{100} \small 1$

Correct answer:

$\dpi{100} \small 1$

Explanation:

The area of the big square is 5, and area of the small square is 1. Therefore, the area of the four right triangles is $\dpi{100} \small 5-1=4$ .

Since the four triangles are all exactly the same, the area of each of the right triangle is 1. We know that the longer side is 2 times the shorter side, so we can represent the shorter side as $\dpi{100} \small x$ and the longer side as $\dpi{100} \small 2x$ . Then we are able to set up an equation:

$\dpi{100} \small \frac{1}{2}\times x\times 2x=1$ .

Therefore, x, the length of the shortest side of the right triangle, is 1.

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Example Question #1 : Right Triangles

A triangle has sides of length 9, 12, and 16. Which of the following statements is true?

Possible Answers:

The triangle cannot exist.

The triangle is acute and isosceles.

The triangle is acute and scalene.

The triangle is obtuse and isosceles.

The triangle is obtuse and scalene.

Correct answer:

The triangle is obtuse and scalene.

Explanation:

This triangle can exist by the Triangle Inequality, since the sum of the lengths of its shortest two sides exceeds that of the longest side:

9 + 12 = 21 > 16

The sum of the squares of its shortest two sides is less than the square of that of its longest side:

$9^{2}+12^{2} = 81+144= 225 <256=16^{2}$

This makes the triangle obtuse.

Its sides are all of different measure, which makes the triangle scalene as well.

Obtuse and scalene is the correct choice.

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Example Question #10 : Right Triangles

Using the following right traingle, calculate the value of

(Not drawn to scale.)

Possible Answers:

$1\textup{ cm}$

$6.4\textup{ cm}$

$3\textup{ cm}$

$9\textup{ cm}$

Correct answer:

$3\textup{ cm}$

Explanation:

We can determine the length of the side by using the Pythagorean Theorem:

c^2=a^2+b^2

where c=5, a=4

Our equation is then:

5^2=x^2+4^2

x^2=25-16=9

x=3

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Example Question #1 : Calculating The Length Of The Side Of A Right Triangle

Calculate the length of the side of the following right triangle.

(Not drawn to scale.)

Possible Answers:

$16\textup{ cm}$

$8.94\textup{ cm}$

$20\textup{ cm}$

$14.42 \textup{ cm}$

Correct answer:

$8.94\textup{ cm}$

Explanation:

We can calculate the length of the side by using the pythagorean theorem: c^2=a^2+b^2

where our values are c=12, a=8

we can then solve for :

12^2=8^2+x^2

x^2=12^2-8^2=80

x=8.94

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Example Question #2 : Calculating The Length Of The Side Of A Right Triangle

A right triangle has a hypotenuse of 13 and a height of 5. What is the length of the third side of the triangle?

Possible Answers:

Correct answer:

Explanation:

In order to find the length of the third side, we need to use the Pythagorean theorem. The hypotenuse, c, is 13, and the height, a, is 5, so we can simply plug in these values and solve for b, the length of the base of the right triangle:

a^2+b^2=c^2

b^2=c^2-a^2=13^2-5^2=169-25

$b^2=144\rightarrow b=12$

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Example Question #3 : Calculating The Length Of The Side Of A Right Triangle

The hypotenuse of a $\textup{30-60-90}$ triangle has length $4^{t}$ . Which of the following is equal to the length of its shortest leg?

Possible Answers:

$4 ^{t-1}$

$4 ^{t-2}$

$4 ^{\frac{1}{2} t }$

$4 ^{\frac{1}{2} t-\frac{1}{2}}$

$4 ^{t-\frac{1}{2}}$

Correct answer:

$4 ^{t-\frac{1}{2}}$

Explanation:

The shortest leg of a $\textup{30-60-90}$ triangle is one half the length of its hypotenuse. In this triangle, it is

$\frac{1}{2} \cdot 4^{t} = \frac{4^{t}}{2} = \frac{4^{t}}{ \sqrt{4}} = \frac{4^{t}}{4^{\frac{1}{2}} } = 4 ^{t-\frac{1}{2}}$

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GMAT Math : Calculating the length of the side of a right triangle

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #7 : Right Triangles

Example Question #1 : Calculating The Length Of The Side Of A Right Triangle

Example Question #1 : Right Triangles

Example Question #10 : Right Triangles

Example Question #1 : Calculating The Length Of The Side Of A Right Triangle

Example Question #2 : Calculating The Length Of The Side Of A Right Triangle

Example Question #3 : Calculating The Length Of The Side Of A Right Triangle

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