The given figure shows a trapezium in which AB || DC and diagonal AC and BD intersect at point P. If AP : CP = 3:5 and area of $△ A P B$ is $27 m^{2}$ , then the area of $△ C P D$ is ___.

$72 m^{2}$

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$75 m^{2}$

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$81 m^{2}$

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$84 m^{2}$

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Solution

The correct option is B
$75 m^{2}$

In $Δ A P B$ and $Δ C P D$ ,

$∠ A P B = ∠ C P D$ (vertically opposite angles)

$∠ P D C = ∠ P B A$ (alternate angles)

$∴ Δ A P B \sim Δ C P D$ (by AA similarity criterion)

$\frac{A P^{2}}{C P^{2}} = \frac{a r (Δ A P B)}{a r (Δ C P D)}$

(Since the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.)

${(\frac{A P}{C P})}^{2} = \frac{a r (Δ A P B)}{a r (Δ C P D)}$

$\Rightarrow {(\frac{3}{5})}^{2} = \frac{27 m^{2}}{a r (Δ C P D)}$

$\Rightarrow a r (Δ C P D) = \frac{27 \times 25}{9} = 75 m^{2}$

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Similar questions

The given figure shows a trapezium in which AB || DC and diagonal AC and BD intersect at point P. If AP : CP = 3:5 and area of $△ A P B$ is $27 m^{2}$ , Find the area of $△ C P D$ .

Q. Diagonal AC and BD of trapezium ABCD, in which AB || DC, intersect each other at O. The triangle which is equal in area of ΔAOD is

(a) ΔAOB

(b) ΔBOC

(c) ΔDOC

(d) ΔADC

Q. In the given figure, ABCD is a trapezium in which AB || DC and diagonals AC and BD intersect at O. Prove that ar(∆AOD) = ar(∆BOC).

The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD intersect at point P. If AP : CP = 2 : 3,

$A r (Δ C P D) : A r (Δ A P B) = ?$

State true or false:

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P, then

Δ A P B

is similar to

Δ C P D .

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The given figure shows a trapezium in which AB || DC and diagonal AC and BD intersect at point P. If AP : CP = 3:5 and area of △APB is 27 m2, then the area of △CPD is ___.

The given figure shows a trapezium in which AB || DC and diagonal AC and BD intersect at point P. If AP : CP = 3:5 and area of $△ A P B$ is $27 m^{2}$ , then the area of $△ C P D$ is ___.