The bisectors of adjacent angles A and B of the parallelogram ABCD

The bisectors of adjacent angles A and B of the parallelogram ABCD intersect the sides BC and AD at points M and N. Find the perimeter of the parallelogram if MC = 3 m, AN = 8m

It is known that the bisector of the angle of a parallelogram cuts off an isosceles triangle from the parallelogram.

This means that triangles ABN and ABM are isosceles.

From triangle ABN, side AB = AN, since angle ABN = ANB.

Hence, AB = 8 m.

From triangle ABM, side AB = BM, since angle BAM = BMA.

Hence, BM = 8 m.

In a parallelogram, opposite sides are parallel and equal.

Hence, BC = AD, AB = CD = 8 m.

BC = BM + MC = 8 + 3 = 11 m.

Then, the perimeter of the parallelogram is P = 2 * (AB + BC) = 2 * (8 + 11) = 38 m.