Mathematics
In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB // DC, prove that :
(i) Area of (△ DOC) = Area of (△ AOB)
(ii) Area of (△ DCB) = Area of (△ ACB)
(iii) ABCD is a parallelogram.
Theorems on Area
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Answer
(i) In △ DOC and △ AOB,
⇒ ∠DOC = ∠AOB (Vertically opposite angles are equal)
⇒ OD = OB (Given)
⇒ ∠DCO = ∠OAB (Alternate angles are equal)
∴ △ DOC ≅ △ AOB (By A.A.S. axiom)
We know that,
Area of congruent triangles are equal.
∴ Area of (△ DOC) = Area of (△ AOB).
Hence, proved that area of (△ DOC) = area of (△ AOB).
(ii) From part (i),
⇒ Area of (△ DOC) = Area of (△ AOB)
⇒ Area of (△ DOC) + Area of (△ BOC) = Area of (△ AOB) + Area of (△ BOC)
⇒ Area of (△ DCB) = Area of (△ ACB).
Hence, proved that area of (△ DCB) = area of (△ ACB).
(iii) We know that,
Area of triangles on same base and between same parallel lines are equal.
Triangles DCB and ACB lie on same base BC and are equal in area.
∴ They lie between same parallel lines.
∴ AD // BC
Also,
AB // DC (Given)
Since, both pairs of opposite sides are parallel,
∴ ABCD is a parallelogram.
Hence, proved that ABCD is a parallelogram.
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