# Ex.12.1 Q2 Areas Related to Circles Solution - NCERT Maths Class 10

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## Question

The radii of two circles are $$8\,\rm{cm}$$ and $$6\,\rm{cm}$$ respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Video Solution
Areas Related To Circles
Ex 12.1 | Question 2

## Text Solution

What is known?

What is unknown?

Radius of $$3^\rm{rd}$$ circle.

Reasoning:

Using the formula of area of circle $$A = \pi {r^2}$$ we find the radius of the circle.

Steps:

Radius of $$(r_1)$$ $$1^\rm{st}$$ circle $$= 8\,\rm{cm}$$

Radius of $$(r_2)$$$$2^\rm{nd}$$ circle $$= 6\,\rm{cm}$$

Let the radius of $$3^\rm{rd}$$ circle $$=r.$$

Area of

$$1^\rm{st}$$circle $$=\pi \rm{r}_{1}^{2}= \pi (8)^2= 64\pi$$

Area of

$$2^\rm{nd}$$ circle $$=\pi \rm{r}_{2}^{2}= \pi (6)^2= 36\pi$$

Given that,

Area of $$3^\rm{rd}$$ circle $$=$$ Area of $$1^\rm{st}$$  circle $$+$$ Area of $$2^\rm{nd}$$ circle

\begin{align}{\pi{{{r}}^{{2}}}}& = {\pi {{r}}_{{1}}^{{2}}\,{{ + }}\,\pi {{r}}_{{2}}^{{2}}}\\{\pi {{{r}}^{{2}}}} &= {64\pi \,{{ + }}\,36\pi }\\{\pi{{{r}}^{{2}}}} &= {{100\pi }}\\{\,\,{{{r}}^{{2}}}} &={{{100}}}\\{\,\,{{r}}} &= \,\pm \,10\end{align}

However, the radius cannot be negative. Therefore, the radius of the circle having area equal to the sum of the areas of the two circles is $$10\,\rm{cm.}$$

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