Isosceles Reduction
Age
14 to 16
Challenge level
Weekly Problem 29 - 2010
An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?
Problem
$PQR$ is a triangle and $S$ is a point on $QR$.
$QP=QR=9$ cm and $PR = PS =6$ cm.
What is the length of $SR$?
Image
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Triangles $PRS$ and $QPR$ are similar because $\angle PSR = \angle QRP$ (since $PR =PS$) and $\angle PRS = \angle QPR$ (since $QP =QR$).
Hence $\frac{SR}{RP} = \frac{RP}{PQ}$, that is $\frac{SR}{6} = \frac{6}{9}$, that is $SR = 4$.