In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2. DP produced meets AB produced at Q. Given the area of triangle CPQ = 20 cm².
Calculate :
(i) area of triangle CDP,
(ii) area of parallelogram ABCD.

Question

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2. DP produced meets AB produced at Q. Given the area of triangle CPQ = 20 cm².
Calculate :
(i) area of triangle CDP,
(ii) area of parallelogram ABCD.

KnowledgeBoat · Accepted Answer

(i) In △BPQ and △CPD, ⇒ ∠BPQ = ∠CPD [Vertically opposite angles are equal] ⇒ ∠BQP = ∠PDC [Alternate angles are equal] ∴ △BPQ ~ △CPD [By AA] Since, corresponding sides of similar triangle are proportional to each other. . As ΔBPQ and ΔCPQ have common vertex at Q and their bases BP and CP are along the same straight line. So, We know that, The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. Hence, area of ΔCPD = 40 cm². (ii) From part (i), Let PQ = x and PD = 2x. From figure, QD = PQ + PD = x + 2x = 3x. ........(1) In △BPQ and △AQD, ⇒ ∠QBP = ∠QAD [Corresponding angles are equal] ⇒ ∠BQP = ∠AQD [Common] ∴ △BPQ ~ △AQD [By AA] Since, corresponding sides of similar triangle are proportional to each other. We know that, The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. Area of trapezium ADPB = Area of △AQD - Area of △BPQ = 90 - 10 = 80 cm². Area of || gm ABCD = Area of △CDP + Area of trapezium ADPB = 40 + 80 = 120 cm². Hence, area of || gm ABCD = 120 cm².

Mathematics

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2. DP produced meets AB produced at Q. Given the area of triangle CPQ = 20 cm².
Calculate :
(i) area of triangle CDP,
(ii) area of parallelogram ABCD.

Similarity

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Mathematics

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2. DP produced meets AB produced at Q. Given the area of triangle CPQ = 20 cm2.Calculate :(i) area of triangle CDP,(ii) area of parallelogram ABCD.

Similarity

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