In parallelogram ABCD, side BC is 6 times larger than side AB. Bisectors of angles DAB and ABC intersect line CD

| August 16, 2021 | education

In parallelogram ABCD, side BC is 6 times larger than side AB. Bisectors of angles DAB and ABC intersect line CD at points M and N, respectively. Find the perimeter of parallelogram ABCD if MN = 22.

Since AM is the bisector of angle A, then the angle BAM = MAD, the angle BKA = KAD, both lying crosswise at the intersection of parallel straight lines BC and AD secant AK.

Angle MKC = BKA as vertical angles. Angle CMR = BАК as criss-crossing at the intersection of parallel AB and MN secant AM. Then the triangle KCM is isosceles KC = MC.

Similarly, triangle DPN is isosceles DP = DN.

Triangles ABK and BAP are isosceles, then AB = BK = AP, when and KC = PD, and therefore:

PD = DN = KC = MC.

By condition, BC = 6 * AB, and MN = 22 cm.

Then KS = DP = 5 * AB, which means DN = MC = 5 * AB.

Then MN = DN + CD + MC = 5 * AB + AB + 5 * AB = 22.

11 * AB = 22.

AB = 22/11 = 2 cm.

BC = 2 * 6 = 12 cm.

Then P = 2 * (2 + 12) = 28 cm.

Answer: The perimeter of the parallelogram is 28 cm.



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