Mathematics
In the following figure, BD is parallel to CA, E is mid-point of CA and BD = CA.
Prove that : ar.(△ ABC) = 2 × ar.(△ DBC)
Theorems on Area
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Answer
Since,
⇒ BD || CA
∴ BD || CE
Also,
BD = CE.
Since, BD = CE and BD || CE,
∴ BCED is a parallelogram.
We know that,
The area of triangle on same base and between the same parallels are equal in area.
△ DBC and △ EBC lie on the same base BC and between same parallel lines BC and ED.
∴ Area of △ DBC = Area of △ EBC …………(1)
In △ ABC,
E is the mid-point of AC.
∴ BE is the median of triangle.
∴ Area of △ EBC = Area of △ ABE ……….(2)
From figure,
⇒ Area of △ ABC = Area of △ EBC + Area of △ ABE
⇒ Area of △ ABC = Area of △ EBC + Area of △ EBC [From equation (2)]
⇒ Area of △ ABC = 2 Area of △ EBC
⇒ Area of △ ABC = 2 Area of △ DBC. [From equation (1)]
Hence, proved that area of △ ABC = 2 area of △ DBC.
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