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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

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

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

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 .

Is there an error in this question or solution?

#### APPEARS IN

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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