Calculating the length of the side of a right triangle

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GMAT Quantitative › Calculating the length of the side of a right triangle

Questions 1 - 7

1

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

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:

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

2

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?

$\dpi{100} \small 1$

$\dpi{100} \small 2$

$\dpi{100} \small 3$

$\dpi{100} \small 4$

$\dpi{100} \small 5$

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.

3

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

Explanation

We need to use the Pythagorean theorem:

4

Using the following right traingle, calculate the value of

(Not drawn to scale.)

$3\textup{ cm}$

$1\textup{ cm}$

$9\textup{ cm}$

$6.4\textup{ cm}$

Explanation

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

where

Our equation is then:

5

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

(Not drawn to scale.)

$8.94\textup{ cm}$

$14.42 \textup{ cm}$

$16\textup{ cm}$

$20\textup{ cm}$

Explanation

We can calculate the length of the side by using the pythagorean theorem:

where our values are

we can then solve for :

6

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

The triangle is obtuse and scalene.

The triangle is obtuse and isosceles.

The triangle is acute and scalene.

The triangle is acute and isosceles.

The triangle cannot exist.

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:

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.

7

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?

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

$4 ^{t-1}$

$4 ^{t-2}$

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

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

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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