In a triangle, $P, Q$ 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 quadrilateral $ARPQ$.

Given:

In a triangle, $P, Q$ and $R$ are the mid-points of sides $BC, CA$ and $AB$ respectively.

$AC = 21\ cm, BC = 29\ cm$ and $AB = 30\ cm$.

To do:

We have to find the perimeter of the quadrilateral $ARPQ$.

Solution:

$P, Q, R$ and the mid points of sides $BC, CA$ and $AB$ respectively.

This implies,

$PQ \parallel AB$ and $PQ = \frac{1}{2}AB$

$=\frac{1}{2} \times 30^{\circ}$

$=15 \mathrm{~cm}$

Similarly,

$QR \parallel BC$ and $\mathrm{QR}=\frac{1}{2} \times \mathrm{BC}$

$=\frac{1}{2} \times 29$

$=14.5 \mathrm{~cm}$

$\mathrm{RP} \parallel \mathrm{AC}$ and $\mathrm{RP}=\frac{1}{2} \mathrm{AC}$

$=\frac{1}{2} \times 21$

$=10.5 \mathrm{~cm}$

Perimeter of quadrilateral $ARPQ=A R+R P+P Q+A Q$

$=\frac{1}{2} A B+\frac{1}{2} A C+\frac{1}{2} A B+\frac{1}{2} A C$

$=A B+A C$

$=30+21$

$=51 \mathrm{~cm}$

The perimeter of the quadrilateral $ARPQ$ is $51\ cm$.

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Updated on: 10-Oct-2022

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