Question

In a triangle $PQR$, the medians $PX$ and $RY$ intersect at the centroid $G$. Given $PG=8cm$, $RY=9cm$, and $QR=5cm$, find (i) $GX$, (ii) $PX$, (iii) $GR$, (iv) $GY$ and (v) the perimeter of triangle $GXR$.

Answer 1

We know $PG=8cm$. Since $\frac{PG}{GX}=2$, we get
$GX=\frac{1}{2}PG=\frac{1}{2}\times 8=4cm$.
(ii) $PX=PG+GX=8+4=12cm$.
(iii) We are given $RY=9cm$. We know that $G$ divides $RY$ in the ratio $2:1$. Hence $GR=\frac{2}{3}RY=\frac{2}{3}\times 9=6cm$.
(iv) $GY=RY-GR=9-6=3cm$.
(v) Since $X$ is the mid-point of $QR$, we get $XR=\frac{1}{2}\times QR=\frac{1}{2}\times 5=2.5cm$.

Hence the perimeter of $GXR$ is

$RG+RX+GX=6+2.5+4=12.5cm$.