In an isosceles trapezoid, the diagonals are perpendicular. The height of the trapezoid is 15. Find its midline.
| April 30, 2021 | education
We will construct the height of the CК through the point O of the intersection of the diagonals.
Triangles AOD and BOC are rectangular, since the diagonals intersect at right angles.
In an isosceles trapezoid, the diagonals are equal and are divided into equal segments at the point of intersection. AO = DO, BO = CO, then triangles AOD and BOC are rectangular and isosceles, and the angle OAD = OBC = 45. Then OH and OK are heights and medians, and ВK = CK = BC / 2, AH = DH = AD / 2.
Right-angled triangle AOН isosceles, OH = AH = AD / 2
Right-angled triangle KOВ is isosceles, OK = BK = BC / 2 ..
OH + OK = KН = AD / 2 + BC / 2 = (AD + BC) / 2 = 15 cm.
(AD + BC) / 2 = KM = 15 cm.
Answer: The middle line of the trapezoid is 15 cm.
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