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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 $$ r_{3}= 3.5 cm$$

1797307_1833451_ans_38813098a24e4d988585d00153625875.png

Maths

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