Question

If each diagonal of a quadrilateral separates it into two triangles of equal area, prove that the quadrilateral is a parallelogram.


plzz answer my question
rl arora ch 9 ex-1 q.9

Answer 1

Solution:

The reference image is attached below.

Given ABCD is a quadrilateral.

AC and BD are diagonals of the quadrilateral.

ar(∆ PQS) = ar(∆ QSR)  and

ar(∆ PRQ) = ar(∆ PSR)  

ar(∆ PQS) + ar(∆ QSR) = ar(quadrilateral ABCD)

2 ar(PQS) = ar(quadrilateral ABCD) – – – – – – (1)

$\text{ar}(\Delta P Q S)=\frac{1}{2} \text{ar}(\text {quadrilateral } A B C D)$ – – – – – – (2)

Similarly, $ar(\Delta P S R)=\frac{1}{2} ar(\text {quadrilateral } A B C D)$

From (1) and (2),  

ar(∆ PQS)=ar(∆ PSR)    

∆ PQS and ∆ PSR are on the same base PS.

Therefore, altitude from Q of the ∆ PQS = altitude from R of the ∆ PSR

⇒ PS parallel to QR.

Similarly, PQ parallel to RS.

This implies that opposite sides of the quadrilateral are parallel.

Hence Quadrilateral PQRS is a parallelogram.