Question

In $\Delta PQR$,$PQ=6$ cm, $PR=9$ cm and $M$ is a point on $QR$ such that it divides $QR$ in the ratio $1:2$ and $PM\perp QR$. Find the length of $QR$.

• A. $\sqrt{15}$
• B. $2\sqrt{15}$
• C. $3\sqrt{15}$
• D. $4\sqrt{15}$

Answer 1

Answer: (C)

$3\sqrt{15}$

Let, $QM=x$, then $MR=2x$
In right $\Delta PQM$
${\left(PQ\right)}^2={\left(PM\right)}^2+{\left(QM\right)}^2$
$\Rightarrow 6^2={\left(PM\right)}^2+x^2$
$\Rightarrow {\left(PM\right)}^2=36-x^2⟶1$
In right $\Delta PMR$
${\left(PR\right)}^2={\left(PM\right)}^2+{\left(MR\right)}^2$
$\Rightarrow 9^2=36-x^2+{\left(2x\right)}^2$ [from 1]
$\Rightarrow 81=36-x^2+4x^2$
$\Rightarrow 81-36=3x^2$
$\Rightarrow 3x^2=45$
$\Rightarrow x^2=15$
$\Rightarrow x=\pm \sqrt{15}$
Length of side cannot be negative
$\therefore x=\sqrt{15}$
Therefore, $R=x+2x=3x$ $=3\sqrt{15}$