Question

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

Answer 1

The side of the square is `5sqrt(2)`

Explanation:

Start from a square `ABCD`. Now consider the triangle `ABC`. 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(AB)^2 + overline(BC)^2 = overline(AC)^2`

But `AB` and `BC` are both sides of the square, and then they both equal a certain quantity `l`, while `AC` is the diagonal and equals `10` cm. The previous identity becomes

`2d^2 = 100 \implies d^2 = 50 \implies d=sqrt(50)`

If you like it, you can simplify `sqrt(50)` into `sqrt(25*2)=sqrt(25)*sqrt(2)=5sqrt(2)`.