# Thread: Inscribed square in triangle

1. ## 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.

2. Hello I-Think
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:

$\tfrac34x + x + \tfrac43x = 5$

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

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