Three circles, the radii of which are 2, 3, 4, respectively, are in pairs tangent externally at points A, B, C.

| March 25, 2021 | education

Three circles, the radii of which are 2, 3, 4, respectively, are in pairs tangent externally at points A, B, C. Find the radius of the circle circumscribed about the triangle ABC.

We connect the centers of the circle and determine the lengths of the sides of the formed triangle ABC.

The length of each side of the triangle is equal to the sum of the radii of the two tangent circles.

AC = 4 + 2 = 6 cm.

AB = 4 +3 = 7 cm.

BC = 3 + 2 = 5 cm.

Let’s define the semiperimeter of the triangle ABC.

p = (6 + 7 + 5) / 2 = 9 cm.

By Heron’s theorem, we determine the area of the triangle ABC.

Savs = √p * (p – AB) * (p – AC) * (p – BC) = √9 * 2 * 3 * 4 = √216 = 6 * √6 cm2.

Let us determine the radius of the circle described around the triangle ABC.

R = S / p = 6 * √6 / 9 = 2 * √6 / 3 cm.

Answer: The radius of the circumscribed circle is 2 * √6 / 3 cm.



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