In an isosceles trapezoid ABCD, the lengths of the bases BC and AD are 2 and 5, respectively. Point E is the middle of AD

| April 11, 2021 | education

In an isosceles trapezoid ABCD, the lengths of the bases BC and AD are 2 and 5, respectively. Point E is the middle of AD. The segments BE and CE meet the diagonals AC and BD at points M and N. Find the length of the segment MN.

Consider triangles AME and BMC. In triangles, the angle M is common, the angle MAE = BCM is as criss-crossing angles at the intersection of parallel straight lines BC and AE secant AC, then the triangles are similar in two angles. Segment AE = AD / 2 = 5/2 = 2.5 cm.

Then BC / AE = BM / ME.

BM / ME = 2 / 2.5.

BM = 2 * IU / 2.5.

Segment BE = BM + ME = 2 * ME / 2.5 + ME = 4.5 * ME / 2.5.

Triangles ALL and MNE are also similar in two angles.

Then BE / ME = BC / MN.

MN = BC * ME / BE = 2 * ME / (4.5 * ME / 2.5) = 2 * 2.5 / 4.5 = 5 / 4.5 = 10/9 = 1.11 cm.

Answer: The length of the segment MN is 1.11 cm.



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