In parallelogram ABCD, the bisectors of angles A and B intersect side CD at points M and K, respectively

| June 24, 2021 | education

In parallelogram ABCD, the bisectors of angles A and B intersect side CD at points M and K, respectively, and segments AM and BK intersect at point P. Find the length of side BC if it is known that MK = 6 and AM: AP = 5: 4.

Since AM / AP = 5/4, then AP = 4 * AM / 5. MR = AM – AP = AM – 4 * AM / 5 = AM / 5.

Triangles APB and KPM are similar in two angles, since the angle APB = KPM as vertical angles, the angle ABP = MKP as criss-crossing angles at the intersection of parallel lines AB and KM secant ВK.

Then AB / KM = AP / MP.

AB / 6 = (4 * AM / 5) / (AM / 5.

AB = 6 * 4 = 24 cm.

AD = CM + MK + DC.

Since triangles АDК and ВСМ are isosceles, then АD = DC = ВС = СМ. Then AD = AB = 24 = 2 * BC + KM.

24 = 2 * BC + 6.

2 * BC = 24 – 6 = 18.

BC = 18/2 = 9 cm.

A variant is possible when point P is inside the parallelogram, then:

24 = 2 * BC – 6.

BC = 30/2 = 15 cm.

Answer: The length of the BC side is 9 cm or 15 cm.



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