This is an elementary problem I have been surprisingly finding hard to solve only by using elementary level math (no trigonometry).

The only given parameters are:

**1** is the distance from the intersection (between the diagonal and bisector of $\angle D$) to the side $AB$

**8** is the distance from the intersection (between the diagonal and the bisector) to the side $CB$.

I need to find the length of the side $AB$. How would you solve it, and what's more important (to me) is how would you explain it to a kid who is just starting to learn geometry?

## Best Answer

In the picture below, the red and blue right triangles are similar (because of parallel sides) and the purple triangle $\triangle DOZ$ is an isosceles right triangle (because $DO$ is an angle bisector, making $\angle ODZ = 45^\circ$).

Let $AX = x$. Then $DZ = x$ as well; therefore $OZ = x$, because $\triangle DOZ$ is isosceles; therefore $YC = x$.

Since the red and blue triangles are similar, their legs are in the same ratio: $AX : XO = OY : YC$, or $x : 1 = 8 : x$. Rearranging, we get $x^2 = 8$, or $x=\sqrt 8$.

Finally, $AB = AX + XB = AX + OY = \sqrt8 + 8$.