# Inscribed square in triangle

• May 12th 2009, 11:27 PM
I-Think
Inscribed square in triangle
Good night. I have a geometry problem that's been plaguing me for years. Any hints you all can give would be greatly appreciated.

PQR is a right-angled triangle with PR=3cm and QR=4cm. The square STVU is inscribed in the triangle PQR. What is the length, in cm, of the side of the square?

All your help is much appreciated.
• May 13th 2009, 05:12 AM
Hello I-Think
Quote:

Originally Posted by I-Think
Good night. I have a geometry problem that's been plaguing me for years. Any hints you all can give would be greatly appreciated.

PQR is a right-angled triangle with PR=3cm and QR=4cm. The square STVU is inscribed in the triangle PQR. What is the length, in cm, of the side of the square?

All your help is much appreciated.

All the triangles are similar: 3-4-5.

If $TU = x$ cm, then $ST=UV = x$ cm. So from $\triangle PTS, PT = \tfrac34x$ cm.

And from $\triangle UVQ, UQ = \tfrac43x$ cm.

So, since $PQ = 5$ cm:

$\tfrac34x + x + \tfrac43x = 5$

$\Rightarrow 9x + 12 + 16x = 60$

$\Rightarrow x = \frac{60}{37}$