Question

the radii of two circles are 8cm and 6cm respectively. find the radius of the circle having area equal to the sum of the areas of the two circles​

Answer 1

$\huge\star{\red{\underline{\mathfrak{Answer :-}}}}$

10 cm

$\huge\star{\red{\underline{\mathfrak{Explanation :-}}}}$

Given :-

The radius of first circle = 8 cm

The radius of second circle = 6 cm

The sum of the areas of circle 1 and 2 who is equal to the area of third circle.

To find :-

The radius of third circle.

Solution :-

Area of first circle = $\pi \times {8}^{2}$

Area of second circle = $\pi \times {6}^{2}$

Area of third circle = $\pi \times {r}^{2}$

So, because this sum of the areas of first to circles is equal to the area of third circle.

$\implies(\pi \times {8}^{2} ) + (\pi \times {6}^{2} ) = (\pi \times {r}^{2} )$

$\implies \cancel {\pi }\: ( {8}^{2} + {6}^{2} ) = \cancel {\pi }\times {r}^{2}$

$\implies {8}^{2} + {6}^{2} = {r}^{2}$

$\implies r \: = \sqrt{ {8}^{2} + {6}^{2} }$

$\implies r \: = \sqrt{64 + 36}$

$\implies r = \sqrt{100}$

$\implies r \: = 10 \: cm$

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Formula used :-

Area of a circle = $\pi \times {r}^{2}$

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Answer 2

Given ,

$\star$Radius of first circle (r1) = 8 cm

$\star$Radius of second circle (r2) = 6 cm

$\pink{ \sf \fbox{ \fbox{The \: sum \: of \: area \: of \: first \: and \: second \: circle = area \: of \: third \: circle }}}$

Let , the radius of third circle be r3

According to the question ,

$\sf \hookrightarrow \pi (r_{1} ) (r_{1} ) + \pi (r_{2}) (r_{2}) = \pi (r_{3})(r_{3}) \\ \\ \sf \hookrightarrow \frac{22}{7} \times 64 + \frac{22}{7} \times 36 = \frac{22}{7} \times (r_{3})(r_{3}) \\ \\ \sf \hookrightarrow \frac{1408}{7} + \frac{792}{7} = \frac{22}{7} \times (r_{3})(r_{3}) \\ \\ \sf \hookrightarrow \frac{2200}{7} = \frac{22}{7} \times (r_{3})(r_{3}) \\ \\ \sf \hookrightarrow (r_{3})(r_{3}) = \frac{ \cancel7 \times 2200}{\cancel 7 \times 22} \\ \\ \sf \hookrightarrow (r_{3})(r_{3}) = 100 \\ \\ \sf \hookrightarrow r_{3} = 10 \: \: cm$

Hence , the radius of the third circle is 10 cm respectively