# A diagonal of a square is 10 cm long. How do you find the length of each side of the square?

##### 1 Answer
Oct 13, 2015

The side of the square is $5 \sqrt{2}$

#### Explanation:

Start from a square $A B C D$. Now consider the triangle $A B C$. It's a right triangle, whose catheti are the sides of the square, and whose hypotenuse is the diagonal of the square.

Now, Pythagoras' theorem tells us that

${\overline{A B}}^{2} + {\overline{B C}}^{2} = {\overline{A C}}^{2}$

But $A B$ and $B C$ are both sides of the square, and then they both equal a certain quantity $l$, while $A C$ is the diagonal and equals $10$ cm. The previous identity becomes

$2 {d}^{2} = 100 \setminus \implies {d}^{2} = 50 \setminus \implies d = \sqrt{50}$

If you like it, you can simplify $\sqrt{50}$ into $\sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2}$.