We are given a quadrilateral with two right angles. The lengths of three sides are given as 14, 10, and 5. We need to find the length of the fourth side, labeled as $x$, and express it in simplest radical form.

1. Problem Description

We are given a quadrilateral with two right angles. The lengths of three sides are given as 14, 10, and

5. We need to find the length of the fourth side, labeled as $x$, and express it in simplest radical form.

2. Solution Steps

First, we can divide the quadrilateral into a rectangle and a right triangle. The base of the rectangle is

5. Therefore, the horizontal distance of the right triangle will also be

5. The height of the rectangle is

1

0. Thus, the vertical distance of the right triangle will be 14-10 =

4.

Now we have a right triangle with legs of lengths 4 and 5, and the hypotenuse is the side we want to find, $x$. We use the Pythagorean theorem to solve for $x$:

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

where $a$ and $b$ are the lengths of the legs of the right triangle, and $c$ is the length of the hypotenuse.

In our case, $a = 4$, $b = 5$, and $c = x$. So,

$4^2 + 5^2 = x^2$

$16 + 25 = x^2$

$41 = x^2$

$x = \sqrt{41}$

Since 41 is a prime number, we cannot simplify the square root any further.

3. Final Answer

$\sqrt{41}$