We are given a quadrilateral with sides of length 10, 3, 8 and $x$. There are two right angles, allowing us to decompose the quadrilateral into a rectangle and a right triangle. We want to find the length of the side $x$ to the nearest tenth.

1. Problem Description

We are given a quadrilateral with sides of length 10, 3, 8 and $x$. There are two right angles, allowing us to decompose the quadrilateral into a rectangle and a right triangle. We want to find the length of the side $x$ to the nearest tenth.

2. Solution Steps

First, let's find the lengths of the legs of the right triangle. One leg is the difference between 8 and 3, which is

5. The other leg is

1

0. Then, we use the Pythagorean theorem to find $x$.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

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

In our case, the sides are 5, 10, and $x$. Then we have

$5^2 + 10^2 = x^2$

$25 + 100 = x^2$

$125 = x^2$

$x = \sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}$

Now we approximate the value to the nearest tenth.

$x \approx 5 \cdot 2.236 = 11.180$

Rounding to the nearest tenth, we have $x \approx 11.2$.

3. Final Answer

$x \approx 11.2$