In the rectangle ABCD, points K, P, M, E are the midpoints of its sides. Find the perimeter of the KPME quadrangle if AC = 13.

| February 5, 2021 | education

Since the diagonal divides the rectangle into two equal parts, and KPME is a rhombus with equal sides, the sides of KP and ME are parallel to the diagonal.
Let the side of the rhombus be a.
Consider the segment AL, where L is the point of intersection of the diagonal with the rhombus.
Since the sides of the rhombus KP and ME are parallel to the diagonal, the diagonal divides the side of the rhombus KE into two equal parts, which means that AL is the median.
Then, AL is equal to half the side of the rhombus, that is, AL = a / 2.
On the other hand, KP = LT, L and T are the points of intersection of the rhombus with the diagonal, KP || LT.
KR = a = LT = 13 – 2 * AL = 13 – 2 * a / 2 = 13 – a.
a = 13 – a;
2a = 13.
a = 13/2 = 6.5.
The rhombus perimeter is
P = 4 * a = 4 * 6.5 = 26.



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