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.

Solution:

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.

Proof:

Statement

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)

Reason

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

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