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State whether the following statement is true or false:

If $O$ is any point on the diagonal BD of a parallelogram ABCD. Then we can say that $ar (Δ OAB)$ $= ar (Δ OBC)$ .

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True

False

answer is A.

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Detailed Solution

Given,
In a parallelogram ABCD,

O

is any point on the diagonal B D.
Question Image

Join AC.
Let AC and BD intersect at point P.
Now P is the mid-point of diagonal AC and BD.
Since diagonals of a parallelogram bisect each other, BP is the angle bisector of

∠ A B C

⇒

BP is median of

Δ A B C

∴ a r Δ P A B = a r Δ P C B

… (1) [Median divides the triangle into to equal parts.]
Similarly, OP is the median of

Δ O A C

∴ a r Δ O A P = a r Δ O C P

… (2) [Median divides the triangle into to equal parts.]
Now on adding (1) & (2), we get,

a r Δ P A B + a r Δ O A P = a r Δ O C P + a r Δ P C B ⇒ a r Δ O A B = a r Δ O B C

Thus, it is true that

a r Δ O A B = a r Δ O B C

.
Hence, the correct option is 1.

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