Question

I don't want a simple answer. I want to know how to do it.

Answer 1

The image is a bit hard to make out, so I'm going to make some guesses. It looks like it says $PN=12$, $QP=8$, and $PM=17$, and you're asked to find $QR$.

You know that $\Delta NPM$ and $\Delta NQR$ are similar, which means their corresponding sides are proportional in length. So there's some constant $c$ such that

$NP=cNQ$
$PM=cQR$
$MN=cRN$

To find $QR$, you need to know $PM$ and the constant $c$.

Clearly, $NQ=NP+PQ=12+8=20$. So, the first equation becomes

$NP=cNQ\iff12=20c\implies c=\dfrac{12}{20}=\dfrac35$

Then in the second equation, you have

$17=\dfrac35QR\implies QR=17\times\dfrac53=\dfrac{85}3=28.\overline3$