Finding height of this triangle

Question

Find $QM$?

My question is why can't we apply Pythagoras on $\Delta{QMR}$ as it's a right angle triangle?

Answer 1

You certainly can apply Pythagoras but you only have one side, the $4$. The $8$ is the length of $PR$, not the length of $MR$ as $MR \lt 4$

You are expected to compute the area of the triangle from the given base and altitude, then see the $8$ as a base and $QM$ as an altitude. That will give you $QM$ and then you can use Pythagoras to get $MR$ if you want.

Answer 2

HINT

By similarity of $\triangle PLR$ and $\triangle QMR$ we have

$$\frac{QM}{QR}=\frac{PL}{PR}\implies QM=\frac{PL}{PR}\cdot QR$$

Answer 3

Assuming $\angle PLR$ is a right angle (without which the solution of the problem is not unique), triangle $\triangle PLR$ is similar to triangle $\triangle QMR.$ That is, $$ PR : PL = QR : QM. $$ Substituting the known quantities, $$ 8 : 5 = 4 : QM. $$ Solve for $QM.$