The bisector of angle C of parallelogram ABCD intersects side AD at point M and the extension of side AB

The bisector of angle C of parallelogram ABCD intersects side AD at point M and the extension of side AB beyond point A to point N. Find the perimeter of the parallelogram if AN = 4, DМ = 3.

Let us use the property of the bisector of a parallelogram, which cuts off an isosceles triangle from it. Then the triangle CMD is isosceles, and MD = CD = 3 cm.

The AMN triangle is similar to the CMD triangle and is also equilateral, therefore AH = AM = 4 cm.

Then AD = BC = AM + MD = 4 + 3 = 7cm.

Determine the perimeter of the parallelogram. P = 2 * (AB + BC) = 2 * (7 + 3) = 20 cm2.

Answer: P = 20 cm2.