A rectangle is inscribed in a right-angled triangle, the vertices of which coincide with the midpoints of the sides

| July 5, 2021 | education

A rectangle is inscribed in a right-angled triangle, the vertices of which coincide with the midpoints of the sides of the rectangle. The perimeter of the inscribed quadrilateral is 40. Find the perimeter of the rectangle, knowing that its adjacent sides are 8: 6.

Let the large side of the ABCD rectangle be 8 * X cm, then the smaller side is 6 * X cm.

Points K, M, H, P are the middle of the sides, then the segments KM, MH, HP and KP cut off right triangles equal in two legs from the rectangle. Then КМ = МН = НР = КР, and therefore the quadrangle of the КМНР rhombus.

From the right-angled triangle BKM, we determine the length of the hypotenuse KM.

BK = AB / 2 = 6 * X / 2 = 3 * X cm.

BM = BC / 2 = 8 * X / 2 = 4 * X cm.

KM ^ 2 = ВK ^ 2 + BM ^ 2 = 9 * X ^ 2 + 16 * X ^ 2 = 25 * X ^ 2.

KM = 5 * X cm.

Rkmnr = 20 * X cm.

Then Rkmnr = 40 = 20 * X.

X = 40/20 = 2.

AB = 6 * 2 = 12 cm.

BC = 8 * 2 = 16 cm.

Ravsd = 2 * (12 + 16) = 56 cm.

Answer: The perimeter of the rectangle is 56 cm.



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