Three circles touch each other externally. A triangle is formed when the centers of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.
Circles
4 Likes

Question

Three circles touch each other externally. A triangle is formed when the centers of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.

Circles

4 Likes

KnowledgeBoat · Accepted Answer

Let there be three circles with center A, B and C and ABC be the triangle formed when centres of 3 circles are joined. Let, AB = 6 cm, AC = 8 cm and BC = 9 cm. Let the radii of the circles having centres A, B and C be r₁, r₂ and r₃ respectively. As, ⇒ AB = r₁ + r₂ = 6 ............(1) ⇒ AC = r₁ + r₃ = 8 ............(2) ⇒ BC = r₂ + r₃ = 9 .............(3) Adding all the above equations, we get ⇒ r₁ + r₂ + r₁ + r₃ + r₂ + r₃ = 6 + 8 + 9 ⇒ 2(r₁ + r₂ + r₃) = 23 ⇒ r₁ + r₂ + r₃ = ⇒ r₁ + r₂ + r₃ = 11.5 cm ...........(4) Substituting value of r₂ + r₃ = 9 in equation (4) we get : ⇒ r₁ + 9 = 11.5 ⇒ r₁ = 11.5 - 9 ⇒ r₁ = 2.5 cm. Substituting value of r₁ + r₂ = 6 in equation (4) we get : ⇒ 6 + r₃ = 11.5 ⇒ r₃ = 11.5 - 6 ⇒ r₃ = 5.5 cm. Substituting value of r₁ + r₃ = 8 in equation (4) we get : ⇒ 8 + r₂ = 11.5 ⇒ r₂ = 11.5 - 8 ⇒ r₂ = 3.5 cm. Hence, the radii of the circles are 2.5 cm, 3.5 cm and 5.5 cm.

Mathematics

Three circles touch each other externally. A triangle is formed when the centers of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.

Circles

Related Questions

Two circles touch each other externally at point P. Q is a point on the common tangent through P. Prove that the tangents QA and QB are equal.

If the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus.

Two circles of radii 5 cm and 3 cm are concentric. Calculate the length of a chord of the outer circle which touches the inner.

If the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.