easy geo-metry...

Geometry Level 2

Points P , Q , R and S divide the sides of a rectangle ABCD in the ratio 1 : 2 as shown in the figure. What fraction of area of the rectangle is the area of parallelogram PQRS ?

5 9 \frac{5}{9} 3 5 \frac{3}{5} 2 5 \frac{2}{5} 4 9 \frac{4}{9}

Sakanksha Deo by Sakanksha Deo , 23, India

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1 solution

Let ∣ A B ∣ = x |AB| = x and ∣ A D ∣ = y . |AD| = y.

Then the combined areas of Δ A P S \Delta APS and Δ C R Q \Delta CRQ is ( 1 3 x ) ( 2 3 y ) = 2 9 x y . (\frac{1}{3}x)(\frac{2}{3}y) = \frac{2}{9}xy.

Next, the combined areas of Δ B Q P \Delta BQP and Δ D S R \Delta DSR is ( 2 3 x ) ( 1 3 y ) = 2 9 x y . (\frac{2}{3}x)(\frac{1}{3}y) = \frac{2}{9}xy.

Thus the area of P Q R S PQRS is x y − 2 ∗ 2 9 x y = 5 9 x y xy - 2*\dfrac{2}{9}xy = \dfrac{5}{9}xy , i.e., 5 9 \boxed{\dfrac{5}{9}} the area of rectangle A B C D . ABCD.

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