Solved: "Two circles touch internally. The sum of their..."

Layla-Rose Ellison

Answered question

2022-03-02

Two circles touch internally. The sum of their areas is

116 π c m^{2}

and distance between their centres is 6 cm. Find the radii of the circles

Radich606

Beginner2022-03-03Added 6 answers

Step 1
Given:
Let R and r be the radii of the circles having centres at O and O' respectively
Finding the given
Then,
Sum of areas

= 116 π c m^{2}

π R^{2} + π r^{2} = 116 π

R^{2} + r^{2} = 116

(1)
Given that
Distance between the centres

= 6 c m

O O^{'} = 6 c m

R - r = 6 c m

(2)
Now,

{(R + r)}^{2} + {(R - r)}^{2} = 2 (R^{2} + r^{2})

{(R + r)}^{2} + 36 = 2 \times 116

{(R + r)}^{2} = 2 \times 116 - 36

{(R + r)}^{2} = 232 - 36

{(R + r)}^{2} = 196

R + r = \sqrt{196}

R + r = 14

(3)
Solving (2) and (3)

R - r = 6

\frac{R + r = 14}{2 R = 20}

R = \frac{20}{2}

R = 10

Put R in (3)

R + r = 14

10 + r = 14

r = 14 - 10

r = 4

Therefore, radii of given circles are 10 cm and 4 cm

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