In parallelogram ABCD, points K, L, M, and N are the midpoints of sides AB, BC, CD, and AD, respectively

| September 23, 2021 | education

In parallelogram ABCD, points K, L, M, and N are the midpoints of sides AB, BC, CD, and AD, respectively. Find the perimeter of the quadrilateral KLMN if the diagonal is AC = 20 and the diagonal is BD = 14.

In the construction, 4 pairs of similar triangles can be distinguished: ABC and KBC, ADC and NDC, BCD and LCM, BAD and KAN. In each pair, one triangle has a parallelogram diagonal base, and the other has a line connecting the midpoints of the sides. They are similar, because each triangle from a pair has a common angle, and the sides of the small triangles adjacent to the common angle are exactly half the size of the corresponding sides of the large triangles. The bases of these triangles will also be half the size of the bases of large triangles:
KL = AC / 2;
LM = BD / 2;
MN = AC / 2;
NK = BD / 2.
The perimeter is equal to the sum:
KL + LM + MN + NK = AC / 2 + BD / 2 + AC / 2 + BD / 2 = AC + BD = 20 + 14 = 34.
Answer: 34.



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