In the given figure, ABCD is a ‖gm in which diagonals AC and BD intersect at O. If ar(‖gm ABCD) is 52cm2, then the ar(∆OAB) = ?
Given: ABCD is a ‖gm in which diagonals AC and BD intersect at O and ar(‖gm ABCD) is 52cm2.
Here,
Ar (∆ABD) = ar(∆ABC)
(∵ ΔABD and ΔABC on same base AB and between same parallel lines AB and CD)
Here,
ar(∆ABD) = ar(∆ABC) = 1/2 × ar(||gm ABCD)
(∵ ΔABD and ΔABC on same base AB and between same parallel lines AB and CD are half the area of the parallelogram)
∴ ar(∆ABD) = ar(∆ABC) = 1/2 × 52 = 26cm2
Now, consider ΔABC
Here OB is the median of AC
(∵ diagonals bisect each other in parallelogram)
∴ ar(∆AOB) = ar(∆BOC)
(∵median of a triangle divides it into two triangles of equal area)
ar(∆AOB) = 1/2 × ar(ΔABC)
ar(∆AOB) = 1/2 × 26 = 13cm2
∴ ar(∆AOB) = 13cm2