AD and BC of the base of the trapezoid ABCD, P is the point of intersection of the bisectors of the angles DAB and ABC

| September 11, 2021 | education

AD and BC of the base of the trapezoid ABCD, P is the point of intersection of the bisectors of the angles DAB and ABC, K is the point of intersection of the bisectors of the angles BCD and CDA. Find the length of the midline of the trapezoid if AB = 4cm, CD = 10cm, PK = 1cm.

By the property of the medians of the trapezoid, the medians drawn from the vertices of the lateral sides intersect at right angles, then the triangles ABP and CDK are rectangular. The points of intersection of the medians of the trapezoid, drawn from the tops of the lateral sides, lie on its midline.

Since MH is the middle line of the trapezium, then AM = BM, and CH = DH, then in the triangles ABP and CDK, the segments PM and KH are medians of right-angled triangles, the length of which is half the length of the hypotenuse. PM = AB / 2 = 4/2 = 2 cm, KN = CD / 2 = 10/2 = 5 cm.

Then the middle line MH = PM + PK + KN = 2 + 1 + 5 = 8 cm.

Answer: The length of the middle line is 8 cm.



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