In parallelogram ABCD, the bisector of an acute angle A divides the opposite side BC

In parallelogram ABCD, the bisector of an acute angle A divides the opposite side BC into segments BK = 8 and KC = 5. Find the perm of the parallelogram ABCD.

1) Perimeter ABCD = AB + BC + CD + AD = 2 * (AB + BC).

2) After drawing the bisector of the angle <A, AK, two segments are obtained: BK and KС, BC = BK + KС = (8 + 5) = 13.

3) Consider a triangle ABK, in which the angles at the base of the AK are equal. <BAK = <BKA, since <DAK = <BAK, as the angles obtained by dividing the angle <A by the bisector of AK, and the angles <KAD = <BKA, as angles with parallel lines AD and BC and the secant AK.

4) Hence, the triangle ABK is isosceles, AB = BK = 8.

5) Perimeter P = 2 * (8 + 13) = 2 * 21 = 42.