The perimeter of parallelogram ABCD is 34. The perimeter of triangle ABC is 25. Find the length of the diagonal AC of parallelogram ABCD.

Given:
parallelogram ABCD,
P ABCD = 34,
diagonal AC of parallelogram ABCD,
R ABC = 25.
Find the length of the diagonal AC -?
Decision:
1. Consider a triangle ABC and a triangle CDA. They have sides AD = BC and CD = AB and angle D = angle B as the opposite sides and angles of the parallelogram are equal. Therefore triangle ABC = triangle CDA on two sides and the angle between them.
2. Consider the parallelogram ABCD.
AC = (P ABC + P CDA – P ABCD -): 2;
AC = (25 + 25 – 34): 2;
AC = (50 – 34): 2;
AC = 16: 2;
AC = 8.
Answer: 8.