Mathematics
In △ ABC, D is a point on side AB and E is a point on AC. If DE is parallel to BC, and BE and CD intersect each other at point O; prove that :
(i) area (△ ACD) = area (△ ABE)
(ii) area (△ OBD) = area (△ OCE)
Answer
(i) Given: ABC is a triangle. D is any point on AB and E is a point on AC. DE is parallel to BC, and BE and CD intersect each other at point O.
To prove: area (△ ACD) = area (△ ABE)
Proof: We know that if two triangles are on the same base and between the same parallels, then the area of the triangles is always equal.
∴ Ar.(Δ BDE) = Ar.(Δ DEC)
Add the area of Ar.(△ADE) in both sides,
⇒ Ar.(Δ ADE) + Ar.(Δ BDE) = Ar.(Δ ADE) + Ar.(Δ DEC)
⇒ Ar.(Δ AED) = Ar.(Δ ACD)
Hence, area (△ ACD) = area (△ ABE)
(ii) To prove: area (△ OBD) = area (△ OCE)
Proof: We know that if two triangles are on the same base and between the same parallels, then the area of the triangles is always equal.
∴ Ar.(Δ BCD) = Ar.(Δ BCE)
The diagonals BE and DC intersect at O, dividing the trapezium into smaller triangles. Subtract the area of △EOD, which is common to both △BCD and △BCE, from both sides:
⇒ Ar.(Δ BCD) - Ar.(Δ BOC) = Ar.(Δ BCE) - Ar.(Δ BOC)
⇒ Ar.(Δ BOD) = Ar.(Δ EOC)
Hence, area (△ OBD) = area (△ OCE)
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