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Three Circles Touch Each Other Externally. a Triangle is Formed When the Centres of These Circles Are Joined Together. Find Radii of Circle, If the Sides of the Triangle Formed Are 6 Cm, 8 Cm and 9 Cm - ICSE Class 10 - Mathematics

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ConceptChord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line

Question

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

Solution 1

AB = 6 cm, AC = 8 cm and BC = 9 cm

Let radii of the circles having centers A, B and C be `r_1 , r_2 and r_3` respectively

`r_1 + r_3 =8`

`r_3+r_2=9`

`r_2+r_1=6`

adding

`r_1 + r_3 +r_3 +r_2 + r_2 +r_1 = 8+9+6`

`2(r_1 +r_2 +r_3) =23`

`r_1+r_2+r_3=11.5cm`

`r_1+9=11.5{(since r_2+r_3=9)`

`r_1=2.5cm`

`r_2 +6=11.5(since   r_1 +r_3=6)`

`r_2 = 5.5cm`

`r_3+8=11.5(since    r_2 + r_1=8)`

`r_3=3.5cm`

Hence , `r_1=2.5cm, r_2=5.5cm and r_3=3.5cm .`

Solution 2

AB = 6 cm, AC = 8 cm and BC = 9 cm

Let radii of the circles having centers A, B and C be `r_1 , r_2 and r_3` respectively

`r_1 + r_3 =8`

`r_3+r_2=9`

`r_2+r_1=6`

adding

`r_1 + r_3 +r_3 +r_2 + r_2 +r_1 = 8+9+6`

`2(r_1 +r_2 +r_3) =23`

`r_1+r_2+r_3=11.5cm`

`r_1+9=11.5{(since r_2+r_3=9)`

`r_1=2.5cm`

`r_2 +6=11.5(since   r_1 +r_3=6)`

`r_2 = 5.5cm`

`r_3+8=11.5(since    r_2 + r_1=8)`

`r_3=3.5cm`

Hence , `r_1=2.5cm, r_2=5.5cm and r_3=3.5cm .`

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Solution Three Circles Touch Each Other Externally. a Triangle is Formed When the Centres of These Circles Are Joined Together. Find Radii of Circle, If the Sides of the Triangle Formed Are 6 Cm, 8 Cm and 9 Cm Concept: Chord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line.
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