A circle is inscribed in an isosceles trapezoid with base lengths of 4 and 16 cm. What is its radius (cm)?

| September 27, 2021 | education

Since a circle is inscribed in the trapezoid, the sum of the lengths of the bases of the trapezoid is equal to the sum of the lengths of its lateral sides. AB + CD = BC + AD = 4 + 16 = 20 cm.

Since the trapezium, by condition, is isosceles, then AB = CD = (AB + CD) / 2 = 20/2 = 10 cm.

In an isosceles trapezoid, the height drawn to the larger base divides it into two segments, the smaller of which is equal to the half-difference of the base lengths.

Then AH = (AD – BC) / 2 = (16 – 4) / 2 = 12/2 = 6 cm.

From the right-angled triangle ABH, according to the Pythagorean theorem, we determine the length of the leg BH.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 100 – 36 = 64.

BH = 8 cm.

The radius of the inscribed circle is equal to half the length of the height of the trapezoid.

R = BH / 2 = 8/2 = 4 cm.

Answer: The radius of the inscribed circle is 4 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.