Need a hint to find $x$

Question

In below picture ,I need to find $x$ .

$x$ is side length of square .I have no clue to find the value of $x$ . As honestly as possible ,the question was for my student's and she asked me to find $x$ .I was thinking ,but I am stuck on this problem .thanks in advanced

Answer 1

The key of finding the side length of the square is just the pythagorean theorem.

Answer 2

Here's a solution/sketch:

1) Consider drawing diagonal AD. Let the intersection of AD with BC be called E.

2) Since AB and CD are parallel, angle BAE is the same as CDE. So by AA similarity triangles ABE and CDE are similar.

3) Therefore, $BE=4\cdot\frac{3}{8}=\frac{3}{2}$ and $CE=4\cdot\frac{5}{8}=\frac{5}{2}.$

4) By the Pythagorean Theorem, $AE=\frac{3\sqrt{5}}{2}$ and $DE=\frac{5\sqrt{5}}{2}.$

5) So $AD=4\sqrt{5}$ and $\boxed{x=2\sqrt{10}}.$

Answer 3

$$AD=\sqrt{8^2+4^2}=4\sqrt{5}$$ this is the diagonal of square so $x=\dfrac{4\sqrt{5}}{\sqrt{2}}=2\sqrt{10}$.