IXL Similarity and altitudes in right triangles

Question

If PR=12 and RS=7 what is QR?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.

Answer 1

PS = 5 (by subtraction)

PS / QS = QS / SR

Can you substitute values and answer?

Answer 2

First I suggest that you write out the all the proportions which govern the 3 right triangles involved.

Once you do that you will find this one:

PQ/PS =PR/PQ

PR=PS+SR

Substitution will give you this quadratic:PQ²=PS²+PS*SR

144=PS²+7PS which has only one solution which make sense, namely 9

Then QS=sqrt(144-81)