How to find the length of a side in a right triangle

Question

If the length of a side in a right triangle is 8 and the hypothenuse is $\sqrt{113}$, what is the length of the other side ?

I tried different formulas but I'm not getting any answer.

Answer choices -

A. 7

B. 9

C. $\sqrt{177}$

D. 19

Answer 1

Hint. We are looking for a positive number $a$ such that $$ a^2+8^2=113 $$$$ a^2=49. $$

Answer 2

Use the Pythagorean theorem.

It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

$$a^2+b^2=c^2$$

In a picture, it looks like:

So, in your example:

$$a^2+8^2=\left(\sqrt{113}\right)^2\Longleftrightarrow a^2+64=113\Longleftrightarrow a=\pm\sqrt{113-65}=\pm7$$

But, in a triangle $a$ can only be positive, so $a=7$.