KnowledgeBoat Logo

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

4 Likes

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.

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. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

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 = 232\dfrac{23}{2}

⇒ 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.

Answered By

3 Likes


Related Questions