Question

# Three circles touch  each other externally . A triangle is formed  when the centre s 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

Solution

## Let ABC be the triangle formed when centres of 3 circles are joined . Given , AB = 6 cm , AC = 8 cm and BC = 9 cm  And let the radii of the circles having centers A,B  and C be $$r_{1}, r_{2} and r_{30}$$ respectively . So, we have $$r_{1}+r_{3}=8$$$$r_{3}+r_{2}=9$$$$r_{2}+r_{1}=6$$Adding all the above equations, we get $$r_{1}+r_{3}+r_{3}+r_{2}+r_{2}+r_{1}= 8+9+62(r_{1}+r_{2}+r_{3}) = 23$$ So, $$r_{1},+r_{2}+r_{3}=11.5 cm$$Now, $$r_{1}+9=11.5 (Asr_{2}+r_{3}=9)$$$$r_{1}= 2.5 cm$$ And, $$r_{2}+6 = 11.5 (As r_{1}+r_{2}=6)$$$$r_{2}=5.5 cm$$Lastly $$r_{3}+8 = 11.5 ( As r_{2}+r_{1}=8)$$$$r_{3}=3.5 cm$$ Therefore , the radii of circles are  $$r_{1}= 5.5 cm$$ and $$r_{3}= 3.5 cm$$ Maths

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