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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 enter image description here

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    Hint: Trace out a line from A to D, the side BC wiil be divided in two parts ($y$ and $4-y$). Then use pythagorean theorem to find the diagonal, after that the side is done! – 2017-02-21
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    Maybe of interest http://math.stackexchange.com/questions/1643786/the-area-of-square. – 2017-02-22

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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 $\boxed{x=2\sqrt{10}}.$

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The key of finding the side length of the square is just the pythagorean theorem. enter image description here

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    DANG this is a great solution.. (I didn't understand what MyGlasses was doing until this) – 2017-02-21
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$$AD=\sqrt{8^2+4^2}=4\sqrt{5}$$ this is the diagonal of square so $x=\dfrac{4\sqrt{5}}{\sqrt{2}}=2\sqrt{10}$.

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    $AB +CD=AD$ ? is this right ? Is this what you mean ? – 2017-02-21
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    think about vector type. – 2017-02-21
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    I get your idea ....thank you and +1+1+1... – 2017-02-21