Question

A circle touches the hypotenuse of a right-angled triangle at its midpoint and passes through the midpoint of the shorter side. If $3$ units and $4$ units are the length of the sides other than hypoteuse and $r$ be the radius of the circle, then the value of $3r$ is

Answer 1

Answer: 5

Using coordinate geometry, assume endpoints of hypotenuse as $A(3,0)$ and $B(0,4)$ , the third vertex as origin,
such that the midpoint of the hypotenuse is $P\left(\frac{3}{2},2\right)$ and the midpoint of the shorter side is $Q\left(\frac{3}{2},0\right)$
Let centre be $C(h,k)$
$\because CP\perp AB$
$\Rightarrow \frac{2-k}{\frac{3}{2}-h}=\frac{3}{4}$
$\Rightarrow 6h-8k=-7$ … (i)
$\because CP=CQ$
$\Rightarrow (h-\frac{3}{2}{)}^2+(k-2{)}^2=(h-\frac{3}{2}{)}^2+k^2$
$\Rightarrow k=1$ and
$6h=1\Rightarrow h=\frac{1}{6}$
$\therefore$ radius $(r)=CQ=\sqrt{(\frac{1}{6}-\frac{3}{2}{)}^2+1}=\sqrt{(\frac{1-9}{6}{)}^2+1}=\frac{5}{3}$
$\therefore 3r=5$