In parallelogram ABCD AB = 6 cm, BC = 9 cm. Points K and E lie on sides BC and CD, respectively, so that CK = 6 cm

univerkov | February 23, 2021 | education

In parallelogram ABCD AB = 6 cm, BC = 9 cm. Points K and E lie on sides BC and CD, respectively, so that CK = 6 cm, CE = 4 cm. The segment KE intersects the diagonal AC at point P. Find the ratio AP and PC.

Let’s draw the ВD diagonal, which divided the AC diagonal in half. AO = OВ = AC / 2.

Consider the triangles ВСD and KСE, in which the sides are proportional, and the angle C of the triangles is common, then these triangles are similar with the similarity coefficient K = BC / CK = 9/6 = 3/2.

Then OC / PC = 3/2.

And since OС = AC / 2, then (AC / 2) / PC = 3/2.

RS = AC / 3.

AР = AC – AC / 3 = 2 * AC / 3.

Then AP / PC = (2 * AC / 3) / (AC / 3) = 2/1.

Answer: The ratio AР / PC = 2/1.

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