If Each Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area then Prove that the Quadrilateral is a Parallelogram

Here we will prove that if each diagonal of a quadrilateral divides it in two triangles of equal area then prove that the quadrilateral is a parallelogram.


Given: PQRS is a quadrilateral whose diagonals PR and QS cut at O such that

ar(∆PQR) = ar(∆PSR), and

ar(∆PQS = ar(∆QRS).

To prove: PQRS is a parallelogram.



1. ar(PQR) = ½ ar(quadrilateral PQRS).

2. ar(PSQ) = ½ ar(quadrilateral PQRS).

3. ar(PQR) = ar(PSQ)

4. SR ∥ PQ.

5. QR ∥ PS.

6. PQRS is a parallelogram. (Proved)


1. Given, ar(∆PQR) = ar(∆PSR).

2. Given, ar(∆PSQ) = ar(∆QRS).

3. From statements 1 and 2.

4. ∆PQR and ∆PSQ are of equl area on the same base PQ.

5. Similarly with PS as base.

6. From statements 4 and 5.

9th Grade Math

From Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area to HOME PAGE

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?