An isosceles trapezoid is described around a circle with a radius of 12 dm

| March 20, 2021 | education

An isosceles trapezoid is described around a circle with a radius of 12 dm, the point of contact divides its lateral side in a ratio of 9: 4. Find the midline of the trapezoid.

Let’s build the radii OP, OK, OM, OE.

Let the length of the segment BP = 4 * X dm, then AR = 9 * X cm, AB = 13 * X dm.

By the property of a tangent drawn from one point, ВK = BP = 4 * X dm. Then BC = 2 * VK = 8 * X dm.

Similarly, AD = 2 * AM = 18 * X dm.

Let’s build the height of the ВН which is equal to the diameter of the inscribed circle, ВН = 2 * R = 2 * 12 = 24 dm.

The height BH divides the base AD into two segments. The length AH is equal to the half-difference of the lengths of the trapezoid bases. AH = (AD – BC) / 2 = 10 * X / = 5 * X dm.

In a right-angled triangle ABН, according to the Pythagorean theorem, AB ^ 2 = AH ^ 2 + BH ^ 2.

169 * X ^ 2 = 25 * X ^ 2 + 576.

144 * X ^ 2 = 576.

X ^ 2 = 576/144 = 4.

X = 2.

BC = 8 * 2 = 16 dm.

AD = 18 * 2 = 36 dm.

Then the middle line is equal to: FG = (BC + AD) / 2 = 52/2 = 26 dm.

Answer: The middle line of the trapezoid is 26 dm.



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