Here we will prove that the area of a rhombus is equal to half the product of its diagonals.
Solution:
Given:
PQRS is a rhombus whose diagonals are PR and QS. The diagonals intersect at O.
To prove: ar(rhombus PQRS) = \(\frac{1}{2}\) ×PR × QS.
Statement |
Reason |
1. ar(∆RSQ) = \(\frac{1}{2}\) ×Base × Altitude = \(\frac{1}{2}\) ×QS × RO. |
1. QS ⊥ PR, because diagonals of a rhombus are perpendicular to each other. |
2. ar(∆PQS) = \(\frac{1}{2}\) ×Base × Altitude = \(\frac{1}{2}\) ×QS × PO. |
2. As in reason 1. |
3. ar(∆RSQ) + ar(∆PQS) = \(\frac{1}{2}\) ×QS × (RO + PO). |
3. By addition from statements 1 and 2. |
4. ar(rhombus PQRS) = \(\frac{1}{2}\) ×PR × QS. |
4. By addition axiom for area. |
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