Question

In the figure $G$ is the centroid of triangle $PQR$. If $MQ=9cm$, $PG=4.6cm$, $KG=2cm$ and $LQ=3.8cm$, write the measurement of the following:
(i) $RG=$ ___________;
(ii) $KR=$ ___________;
(iii) $GL=$ ___________;
(iv) $PL=$ ___________;
(v) $MG=$ ___________;
(vi) $GQ=$ ___________;
(vii) $RL=$ ___________;
(viii) $RQ=$ ___________.

Answer 1

Centroid divides each median in the ratio of $1:2$

$\left(i\right)$Consider the median $RK$. $KG$ and $RG$ are divided in the ratio $1:2$

$\Rightarrow \frac{KG}{RG}=\frac{1}{2}$

We know $KG=2cm$

$\Rightarrow \frac{2}{RG}=\frac{1}{2}$

Cross multiply

$\Rightarrow RG=4cm$

$\left(ii\right)$ $KR=KG+RG$

$KG=2cm$ and $RG=4cm$

$\therefore KR=4+2=6cm$

$\left(iii\right)$Consider the median $PL$. $PG$ and $GL$ are divided in the ratio $1:2$

$\Rightarrow \frac{GL}{PG}=\frac{1}{2}$

We know $PG=4.6cm$

$\Rightarrow \frac{GL}{4.6}=\frac{1}{2}$

Cross multiply

$\Rightarrow GL=2.3cm$

$\left(iv\right)$ $PL=PG+GL$

$PG=4.6cm$ and $GL=2.3cm$

$\therefore PL=4.6+2.3=6.9cm$

$\left(v\right)$ $MQ$ is divided in such a way that it should form a ratio of $1:2$

$MQ=9cm$

$\therefore MG:GQis1:2$

If $9$ has to be divided in a ratio of $1:2$ t it will be $3and6$

$MG$ will be $3cm$ since its the smaller number.

$\left(vi\right)$ $MQ$ is divided in such a way that it should form a ratio of $1:2$

$MQ=9cm$

$\therefore MG:GQis1:2$

If $9$ has to be divided in a ratio of $1:2$ t it will be $3and6$

$GQ$ will be $6cm$ since its the larger number.

$\left(vii\right)$ $PL$ is the median and it divides $RQ$ in the middle.

Therefore $RL=LQ$

$LQ=3.8cm$

$\therefore RL=3.8cm$

$\left(viii\right)$ $RQ=RL+LQ$

We know $RL=LQ=3.8cm$

$RQ=3.8+3.8=7.6cm$