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. |
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