​ help

Use the Pythgorean Thearom and find the hypotenuse remember a^2+b^2=c^2

divide RQ by 2 and divide PQ by 2 then use the Pythgorean Thearom to find ZX

Notice that \(\triangle RYX \sim \triangle XZQ\) by AA postulate.

 

Let x be the side length of the square. Then

\(ZQ = 12-x\\ RY=8-x\\ YX=x\\ XZ=x\)

 

By similar triangles,

\(\dfrac{RY}{YX} = \dfrac{XZ}{ZQ}\\ \dfrac{8-x}{x}=\dfrac{x}{12-x}\)

 

Solving this equation gives the required answer.

 

Ans: Side length = 24/5.

 

As an exercise, you can try to show that when PQ = a, PR = b, the side length of the square is \(\dfrac{ab}{a + b}\).