Question

In $△ABC$, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC= 29 cm and AB =30 cm. Find the perimeter of the quad ARPQ.

• A. $91cm$
• B. $60cm$
• C. $51cm$
• D. $70cm$

Answer 1

The correct option is C $51cm$

$\Rightarrow$ In $△ABC$, $R$ and $P$ are the mid-points of $AB$ and $BC$

$\therefore$ $RP\parallel AC,$ $RP=\frac{1}{2}AC$ [ By mid-point theorem ]

In a quadrilateral $RPQA$,

$\Rightarrow$ $RP\parallel AQ,RP=AQ$

$\therefore$ $RPQA$ is a parallelogram

$\Rightarrow$ $AR=\frac{1}{2}AB$

$\therefore$ $AR=\frac{1}{2}\times 30=15cm$

$\Rightarrow$ $AR=PQ=15cm$ [ Since, opposite sides are equal ]

$\Rightarrow$ $RP=\frac{1}{2}AC=\frac{1}{2}\times 21=10.5cm$ [ Since, opposite sides are equal ]

$\Rightarrow$ Perimeter of $ARPQ=AR+QP+RP+AQ$

$=15+15+10.5+10.5$

$=51cm$

$\therefore$ Perimeter of $ARPQ$ is $51cm$