We are given a figure that can be decomposed into a right isosceles triangle on top of a rectangle. The legs of the right isosceles triangle have length 6. The diagonal of the rectangle has length 10. We need to find the length of the side $x$ of the figure.

1. Problem Description

We are given a figure that can be decomposed into a right isosceles triangle on top of a rectangle. The legs of the right isosceles triangle have length

6. The diagonal of the rectangle has length

1

0. We need to find the length of the side $x$ of the figure.

2. Solution Steps

First, let us find the length of the base of the isosceles right triangle. By the Pythagorean theorem, the length of the base is $b = \sqrt{6^2 + 6^2} = \sqrt{36+36} = \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

This length is also the length of the horizontal side of the rectangle. Let $y$ be the height of the rectangle. By the Pythagorean theorem applied to the rectangle with diagonal of length 10, we have

$y^2 + (6\sqrt{2})^2 = 10^2$

$y^2 + 36 \cdot 2 = 100$

$y^2 + 72 = 100$

$y^2 = 100 - 72$

$y^2 = 28$

$y = \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}$.

The side $x$ consists of the height of the rectangle plus the side of the isosceles right triangle, which is

6. Therefore, $x = y + 6 = 2\sqrt{7} + 6$.

3. Final Answer

$x = 6 + 2\sqrt{7}$