# Three circles, the radii of which are equal to 2, 3 and 4, in pairs are tangent externally.

October 9, 2021 | education

| **Three circles, the radii of which are equal to 2, 3 and 4, in pairs are tangent externally. Find the area of a triangle whose vertices are the centers of the circles.**

The figure shows that the sides of the triangle ABC are equal to the sums of the radii of the contiguous circles. AB = AA1 + A1B = 4 + 3 = 7 cm, AC = AC1 + CC1 = 4 + 2 = 6 cm, CB = BB1 + CB1 = 3 + 2 = 5 cm.

We define the area of a triangle by Heron’s theorem.

Sас = √р * (р – а) * (р – b) * (p – c), where р – half-perimeter of a triangle, а, b, c – sides of a triangle.

p = (AB + BC + CA) / 2 = (7 + 6 + 5) / 2 = 18/2 = 9 cm.

Sav = √9 * (9 – 7) * (9 – 6) * (9 – 5) = √9 * 2 * 3 * 4 = √216 = 6 * √6 cm2.

Answer: The area of the triangle is 6 * √6 cm2.

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