Question

P$(3,7)$, Q$(8,9)$ and R$(11,23)$ are three vertices of the triangle PQR. S divides PQ in the ratio $2:3$ and T divides QR in the ratio $3:2$

Answer 1

Coordinates of S are
$(\frac{16+9}{5},\frac{18+21}{5})=(5,\frac{39}{5})$
Coordinates of T are
$(\frac{33+16}{5},\frac{69+18}{5})=(\frac{49}{5},\frac{87}{5})$
$(ST{)}^2={(5-\frac{49}{5})}^2+{(\frac{39}{5}-\frac{87}{5})}^2$
$={\left(\frac{24}{5}\right)}^2+{\left(\frac{48}{5}\right)}^2=\frac{576}{5}$
$\Rightarrow 5(ST{)}^2=576$
$(PQ{)}^2=29,(QR{)}^2=205,(PR{)}^2=320$
$(PS{)}^2=\left[\right(2/5\left)\right(PQ{)}^2]=\frac{116}{25}$.
$(QS{)}^2=\left[\right(3/5\left)\right(PQ{)}^2]=\frac{261}{25}$.
$(QT{)}^2=\left[\right(3/5\left)\right(QR{)}^2]=\frac{369}{25}$.
So that $(PR{)}^2+(QR{)}^2+51=320+205+51=576$
$100(PS{)}^2=4\times 116=464$
and $(PQ{)}^2+2(QR{)}^2+25=29+410+25=464$
Next, $2(PR{)}^2-64=2\times 320-64=576$ (r)
$50(QS{)}^2-2(PQ{)}^2=2\times 261-2\times 29=464$ (s)
and $5(QT{)}^2+(QR{)}^2+2=369+205+2=576$ (t)
Combining these results we get the required answer.