In $Δ A B C, P Q ∥ B C$ and area of quadrilateral $P B C Q = 42 s q . c m$ . If $A P : P B = 2 : 3,$ then find the area of $Δ A P Q$ .

28 s q . c m

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\frac{56}{3} s q . c m

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8 s q . c m

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33.6 s q . c m

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Solution

The correct option is C $8 s q . c m$
In $△ A B C$ , $P Q ∥ B C$
$∠ B A C = ∠ P A Q$ ...... (Common angle)
$∠ A B C = ∠ A P Q$ ....... (Corresponding angles)
Thus, $△ A B C \sim △ A P Q$
$\Rightarrow \frac{A r (△ A B C)}{A r (△ A P Q)} = \frac{A B^{2}}{A P^{2}}$

$\Rightarrow \frac{A r (△ A P Q) + A r (P B C Q)}{A r (△ A P Q)} = \frac{25}{4}$
$\Rightarrow \frac{x + 42}{x} = \frac{25}{4}$
$\Rightarrow 4 x + 168 = 25 x$
$\Rightarrow 21 x = 168$
$\Rightarrow x = 8$
Thus, $A r (△ A P Q) = 8 s q . c m$

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In ΔABC,PQ∥BC and area of quadrilateral PBCQ=42 sq.cm. If AP:PB=2:3, then find the area of ΔAPQ.

The correct option is C 8 sq.cmIn △ABC, PQ∥BC∠BAC=∠PAQ ...... (Common angle)∠ABC=∠APQ ....... (Corresponding angles)Thus, △ABC∼△APQ⇒Ar(△ABC)Ar(△APQ)=AB2AP2⇒Ar(△APQ)+Ar(PBCQ)Ar(△APQ)=254⇒x+42x=254⇒4x+168=25x⇒21x=168⇒x=8Thus, Ar(△APQ)=8 sq.cm

In $Δ A B C, P Q ∥ B C$ and area of quadrilateral $P B C Q = 42 s q . c m$ . If $A P : P B = 2 : 3,$ then find the area of $Δ A P Q$ .