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Area of a Triangle

i P and Q are...

Question

Question 7 (i)
P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that:
$(i) a r (P R Q) = \frac{1}{2} a r (A R C)$

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Solution

Construction: Join PQ, RQ and RC.

QP and QR medians of triangles QAB and QAP.
$∴ a r (Δ Q A P) = a r (Δ P B Q)$ ------- (1)
And
$∴ a r (Δ Q A P) = 2 a r (Δ P Q R)$ --------- (2)
From (1) and (2), we get,
$\Rightarrow a r (Δ P B Q) = 2 a r (Δ P Q R)$ ------- (3)
$\Rightarrow a r (Δ A R C) = 2 a r (Δ P Q R)$ ------ (4)
$\Rightarrow a r (Δ P Q R) = \frac{1}{2} a r (Δ A R C)$
Hence,
$a r (Δ P Q R) = \frac{1}{2} a r (Δ A R C)$

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Area of a Triangle

Standard IX Mathematics

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