# A circle of radius 6 is inscribed in an isosceles trapezoid. If the point of tangency divides the lateral side into segments

A circle of radius 6 is inscribed in an isosceles trapezoid. If the point of tangency divides the lateral side into segments, the difference between which is 5, then the middle line of the trapezoid is …

Let’s draw from point O the radii OK, OP and OM to the points of tangency. By the property of tangents to the circle drawn from one point ВK = ВM, AK = AP.

Let’s draw the height of the HВ trapezoid. Quadrilateral BMRН is a rectangle, then BM = HP.

By condition, AK – KB = 5 cm.Let AK = X cm, then KB = BM = HP = X – 5 cm.

Then AH = X – (X – 5) = 5 cm.

The height of the trapezoid is equal to two radii of the inscribed circle, then BH = 2 * OM = 2 * 6 = 12 cm.

In a right-angled triangle ABН, according to the Pythagorean theorem, we determine the length of the hypotenuse AB.

AB ^ 2 = BH ^ 2 + AH ^ 2 = 144 + 25 = 169.

AB = 13 cm.

Since, by the property of the tangent, ВН = ВK, and ВK = AР, then AB = ВK + AР.

ВK = BC / 2, AR = BP / 2, then AB = (BC + BP) / 2, which is the middle line of the trapezoid.

NL = AD = 13 cm.

Answer: The middle line of the trapezoid is 13 cm.

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