A straight line parallel to the bases is drawn through the center of the circle inscribed in the trapezoid.

| September 4, 2021 | education

A straight line parallel to the bases is drawn through the center of the circle inscribed in the trapezoid. Prove that the segment of this line, enclosed between the sides, is equal to a quarter of the perimeter of the trapezoid.

Since a circle is inscribed in the trapezoid, the sum of the lengths of the lateral sides is equal to the sum of the lengths of the bases of the trapezoid.

(AB + CD) = (BC + AD).

The KM segment, according to the condition, is drawn through the center of the circle, and since the height of the PH, equal to the diameter of the circle, is halved by the point O, the KM segment is the middle line of the trapezoid.

Then KM = (BC + AD) / 2, and since AB + CD = BC + CD, then

KM = (AB + CD + BC + CD) / 4 = Ravsd / 4, which was required to prove.



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