In a trapezoid inscribed in a circle, you can inscribe a circle with a radius of 4 cm.Find the sides of the trapezoid

| February 17, 2021 | education

In a trapezoid inscribed in a circle, you can inscribe a circle with a radius of 4 cm.Find the sides of the trapezoid if the middle line is 10.

If a trapezoid is inscribed in a circle, then such a trapezoid is isosceles, AB = CD.

Since a circle is also inscribed in a trapezoid, then by the property of such a trapezoid (BC + AD) = (AB + CD).

By condition, the length of the midline of the trapezoid is 10 cm.

MR = (BC + AD) / 2 = 10 cm.

(BC + AD) = 2 * 10 = 20 cm.Then and (AB +CD) = 20 cm, and since AB = CD, then AB = CD = 20/2 = 10 cm.

The height of the trapezoid is equal to the diameter of the inscribed circle, then ВC = 2 * 4 = 8 cm.

From a right-angled triangle ABK, AK ^ 2 = AB ^ 2 – ВK ^ 2 = 100 – 64 = 36. AK = 6 cm.

Since the trapezoid is isosceles, then AK = (AD – BC) / 2.

(AD – BC) = 2 * AK = 2 * 6 = 12 cm.

AD + BC = 20 cm.

AD – BC = 12 cm.

Let’s add two equalities.

2 * AD = 32.

AD = 32/2 = 16 cm.

BC = 20 – 16 = 4 cm.

Answer: The sides of the trapezoid are 10 cm, 4 cm, 10 cm, 16 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.