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in Parallelograms by (58.8k points)

P is any point on the diagonal AC of a parallelogram ABCD. Prove that ar (△ ADP) = ar (△ ABP).

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Best answer

Join the diagonal BD.

From the figure we know that AC and BD are the diagonals intersecting at point O.

We know that the diagonals of a parallelogram bisect each other.

Thus, we get O as the midpoint of AC and BD.

Median of triangle divides it into two triangles having equal area.

Consider △ ABD

We know that OA is the median

So we get

Area of △ AOD = Area of △ AOB ……. (1)

Consider △ BPD

We know that OP is the median

So we get

Area of △ OPD = Area of △ OPB …….. (2)

By adding both the equations we get

Area of △ AOD + Area of △ OPD = Area of △ AOB + Area of △ OPB

So we get

Area of △ ADP = Area of △ ABP

Therefore, it is proved that ar (△ ADP) = ar (△ ABP).

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