Question

in triangle abc p q and R are the midpoints of sides ab bc and ac.if ar(PBQR) is equal to 36 cm square find area of ABC

Answer 1

Answer:

area of triangle ABC is $72cm^2$

Step-by-step explanation:

Given  triangle ABC ,P, Q,and R are the midpoint of sides AB, bc and AC respectively if $ar(PBQR)=36 cm^2$. we have to find the ar(ABC).

In ΔABC, P and Q are the mid points of sides AB and BC ∴ by mid-point theorem PQ is parallel to AC and equal to half of AC.

⇒ PQ=AR and PQ||AR and also PQ=RC and PQ||RC

⇒ QPAR and PQRC both are parallelogram

As One diagonal of a parallelogram divides the parallelogram into 2 congruent triangles of equal area.

⇒ ar(PQR)=ar(APR) and ar(PQR)=ar(QRC)   →   (1)

Similarly, In ΔABC, P and R are the mid points of sides AB and AC  ∴ by mid-point theorem PR is parallel to BC and equal to half of BC

⇒ PR=BQ and PR||BQ

⇒ PRBQ is a parallelogram

As One diagonal of a parallelogram divides the parallelogram into 2 congruent triangles of equal area.

⇒    $ar(PQR)=ar(APR)=\frac{1}{2}ar(PBQR)=18 cm^2$ →  (2)

From eq (1) and (2),  

$ar(APR)=ar(PBQ)=ar(PQR)=ar(QRC)=18cm^2$

$ar(ABC)=4\times ar(PQR)=4\times 18=72cm^2$

Hence area of triangle ABC is $72cm^2$