question_answer

Three circles have the centres at A, B, C and each circle touches the other two externally. If AB = 5 cm, BC = 7 cm and CA = 6 cm, then the radii of three circles respectively are

A)  2, 3, 4                    

B)         3, 4, 5

C)  2, 4, 5                                  

D)  2, 3, 5

Correct Answer: A

Solution :

 Let \[{{r}_{1}},{{r}_{2}},{{r}_{3}}\] be the radii of the circles. Then, from the figure, \[{{r}_{1}}+{{r}_{2}}=5\]                                ?..(i) \[{{r}_{2}}+{{r}_{3}}=7\]                                ?..(ii) \[{{r}_{3}}+{{r}_{1}}=6\]                                ?..(iii) Adding equations (i), (ii) and (Hi) we get \[2({{r}_{1}}+{{r}_{2}}+{{r}_{3}})=18\] or            \[{{r}_{1}}+{{r}_{2}}+{{r}_{3}}=9\]                            ?..(iv) Subtracting respectively equations (ii), (iii), (i) from equation (iv), We get,                                 \[{{r}_{1}}=2,\,\,{{r}_{2}}=3,\,\,{{r}_{3}}=4\]