Question

If $G$ is the centroid and $I$ the incentre of the triangle with vertices $A\equiv (-36,7)$, $B\equiv (20,7)$ and $C\equiv (0,-8)$, then $GI$ is equal to

• A. $\frac{\sqrt{173}}{3}$
• B. $\frac{\sqrt{397}}{3}$
• C. $\frac{\sqrt{205}}{3}$
• D. none of these

Answer 1

Answer: (C)

$\frac{\sqrt{205}}{3}$

We have $a=BC=\sqrt{(20{)}^2+(7+8{)}^2}=25$,
$b=CA=\sqrt{(36{)}^2+(8+7{)}^2}=39$ and
$c=AB=\sqrt{(56{)}^2}=56$
Therefore, the coordinates of $I$ are,
$(\frac{25(-36)+39\left(20\right)+56\left(0\right)}{25+39+56},\frac{25\left(7\right)+39\left(7\right)+56(-8)}{25+39+56})=(-1,0)$
And the coordinates of $G$ are,
$(\frac{-36+20+0}{3},\frac{7+7-8}{3})=(\frac{-16}{3},2)$
$GI=\sqrt{{(-1+\frac{16}{3})}^2+(0-2{)}^2}=\frac{1}{3}\sqrt{205}$