In an isosceles trapezoid, the sum of the bases is 48 cm, and the radius of the inscribed circle is 6√3 cm

| March 2, 2021 | education

In an isosceles trapezoid, the sum of the bases is 48 cm, and the radius of the inscribed circle is 6√3 cm. Find the sides of the trapezoid.

If a circle is inscribed in a trapezoid, then the sum of the sides of the trapezoid is equal to the sum of its bases.

AB + CD = BC + AD = 48.

Since the trapezoid is isosceles, then AB = CD = 48/2 = 24 cm.

From the vertices of the obtuse angles of the trapezoid, we draw the heights of the ВН and СK, the length of which is equal to the diameter of the circle. ВН = СK = 2 * R = 2 * 6 * √3 = 12 * √3 cm.

In a right-angled triangle AСН, according to the Pythagorean theorem, we define the leg AН.

AH ^ 2 = AB ^ 2 – BH ^ 2 = 24 ^ 2 – (12 * √3) ^ 2 = 576 – 432 = 144.

AH = 12 cm.

Since the trapezoid is isosceles, then DK = AH = 12 cm.

Quadrangle НВСК rectangle, then ВС = НК.

Let BC = NK = X cm.

Then BC + НK + AH + DK = 48.

2 * BC = 48 – 12 – 12 = 24 cm.

BC = 24/2 = 12 cm.

Then the base AD = AH + НK + DK = 12 + 12 + 12 = 36 cm.

Answer: The sides are 24 cm, the smaller base is 12 cm, the larger base is 36 cm.



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