Mathematics
P is the mid-point of diagonal AC of quadrilateral ABCD. Prove that the quadrilaterals ABPD and CBPD are equal in area.
Theorems on Area
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Answer
Given: ABCD is a quadrilateral with diagonal AC and P is its mid-point of AC (AP = PC).
To Prove: Area of quadrilateral ABPD = Area of quadrilateral CBPD
Construction: Join PB and DP.
Proof: In Δ ABC, P is the midpoint of AC.
The line segment BP is median. A median divides a triangle into two triangles of equal area.
∴ Ar.(Δ ABP) = Ar.(Δ BCP) …………….(1)
Similarly, in Δ ADC, P is the midpoint of AC.
The line segment DP is median. A median divides a triangle into two triangles of equal area.
∴ Ar.(Δ ADP) = Ar.(Δ DCP) ……………..(2)
Adding equations (1) and (2),
⇒ Ar.(Δ ABP) + Ar.(Δ ADP) = Ar.(Δ BCP) + Ar.(Δ DCP)
⇒ Area of quadrilateral ABPD = Area of quadrilateral CBPD
Hence, the quadrilaterals ABPD and CBPD are equal in area.
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