I4.1 In triangle $A B C, \angle B=90^{\circ}, B C=7$ and $A B=24$. If $r$ is the radius of the inscribed circle, find $r$.
Let $O$ be the centre of the inscribed circle, which touches $B C, C A, A B$ at $P, Q, R$ respectively.
$O P \perp B C, O Q \perp A C, O R \perp A B$ (tangent $\perp$ radius)
$O R B P$ is a rectangle (it has 3 right angles)
$B R=r, B P=r$ (opp. sides of rectangle)
$C P=7-r, A R=24-r$
$A C^{2}=A B^{2}+B C^{2}$ (Pythagoras' Theorem)
$=24^{2}+7^{2}=625$
$A C=25$
$C Q=7-r, A Q=24-r$ (tangent from ext. point)
$C Q+A Q=A C$
$7-r+24-r=25$