We are given a figure consisting of two right triangles sharing a common side. We are given the lengths of two sides of one right triangle (6 and 4), and the length of one side of the other right triangle (8). We are asked to find the length of the side $x$ of the second right triangle.

1. Problem Description

We are given a figure consisting of two right triangles sharing a common side. We are given the lengths of two sides of one right triangle (6 and 4), and the length of one side of the other right triangle (8). We are asked to find the length of the side $x$ of the second right triangle.

2. Solution Steps

First, let's label the shared side as $h$.

For the right triangle on the right, we can use the Pythagorean theorem:

$h^2 + 4^2 = 6^2$

$h^2 + 16 = 36$

$h^2 = 36 - 16$

$h^2 = 20$

$h = \sqrt{20}$

$h = \sqrt{4 \cdot 5}$

$h = 2\sqrt{5}$

Now, for the right triangle on the left, we can also use the Pythagorean theorem:

$x^2 + h^2 = 8^2$

$x^2 + (2\sqrt{5})^2 = 64$

$x^2 + 4 \cdot 5 = 64$

$x^2 + 20 = 64$

$x^2 = 64 - 20$

$x^2 = 44$

$x = \sqrt{44}$

$x = \sqrt{4 \cdot 11}$

$x = 2\sqrt{11}$

3. Final Answer

$x = 2\sqrt{11}$