Choose whether the given statement is true or false.

In Fig. ABCDE is any pentagon. BP drawn parallel to AC meets DC produced at P and EQ drawn parallel to AD meets CD produced at Q. Prove that ar (ABCDE) = ar (ΔAPQ).

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Detailed Solution

Given Points:
1.ABCDE is a pentagon
2.BP || AC
3.EQ || CD
We have to prove that ar (ABCDE) = ar (APO). Where ar stands for area.
So, let’s say that ΔADQ and ΔADE are at the same base that is AD and between the same parallel lines.
Therefore, we can say that their areas are equal, that is ar (ΔADQ) = ar (ΔADE). -----Equation 1
So, let’s say that ΔACP and ΔACB are at the same base that is AC and between the same parallel lines.
Therefore, we can say that their areas are equal, that is ar (ΔACP) = ar (ΔACB). ---Equation 2
Using both the equations 1 and 2, by adding we get,
ar (ΔADQ) + ar (ΔACP) = ar (ΔADE) + ar (ΔACB)
As to make ar (ΔAPQ) on the left side and ar (ABCDE) on right side we must add ar (ΔACD) on both sides of the equation.
Hence, on adding we get,
ar (ΔADQ) + ar (ΔACP) + (ΔACD) = ar (ΔADE) + ar (ΔACB) + (ΔACD)
Therefore, ar (ΔAPQ) = ar (ABCDE)
Hence Proved.and the statement is True.

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