Question

Find x Thanks for the help in advance!

Answer 1

I'm not sure if this is the easiest way of doing this, but it surely work.

Let the base of the triangle be AB, and let CH be the height. Just for reference, we have

$AH=2,\quad HB=6,\quad AC=x$

Moreover, let CH=y and BC=z

Now, AHC, CHB and ABC are all right triangles. If we write the pythagorean theorem for each of them, we have the following system

$\begin{cases}4+y^2=x^2\\36+y^2=z^2\\x^2+z^2=64\end{cases}$

If we solve the first two equations for y squared, we have

$y^2=x^2-4\\y^2=z^2-36$

And we can deduce

$z^2 = x^2+32$

So that the third equation becomes

$x^2+x^2+32=64 \iff 2x^2 = 32 \iff x^2=16 \iff x=4$

(we can't accept the negative root because negative lengths make no sense)