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ABCD is a parallelogram in which diagonals AC and BD intersects at O. If E, F, G, H are mid-points of AO, DO, CO and BO respectively. Then the ratio of the perimeter of the quadrilateral  EFGH to the perimeter of Parallelogram ABCD is

Question

ABCD is a parallelogram in which diagonals AC and BD intersects at O. If E, F, G, H are mid-points of AO, DO, CO and BO respectively. Then the ratio of the perimeter of the quadrilateral  EFGH to the perimeter of Parallelogram ABCD is

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Solution

In a parallelogram, the diagonals bisect each other. Therefore, AO = OC and BO = OD.

Since E, F, G, H are mid-points of AO, DO, CO and BO respectively, we have AE = EO = 1/2 AO, DF = FO = 1/2 DO, CG = GO = 1/2 CO, and BH = HO = 1/2 BO.

Therefore, the lengths of the sides of quadrilateral EFGH are half the lengths of the sides of the parallelogram ABCD.

The perimeter of a polygon is the sum of the lengths of its sides. Therefore, the perimeter of quadrilateral EFGH is half the perimeter of parallelogram ABCD.

So, the ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD is 1:2.

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