Ex.12.1 Q1 Areas Related to Circles Solution - NCERT Maths Class 10

The radii of two circles are \(19\, \rm{cm}\) and \(9\,\rm{cm}\) respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Text Solution

What is known?

Radii of two circles.

What is unknown?

Radius of \(3^\rm{rd}\) circle.

Reasoning:

Using the formula of circumference of circle \({C = 2}\pi r\) we find the radius of the circle.

Steps:

Radius \(({r_1})\)of \(1^ \rm{st}\) circle \(= 19 \,\rm{cm}\)

Radius \(({r_2})\) or \(2^\rm{nd} \) circle \(=9\,\rm{cm}\)

Let the radius of \(3^ \rm{rd}\) circle be \(r.\)

Circumference of \(1^\rm{st}\)circle \(= 2 \pi{{\text{r}}_{\text{1}}}{\text{ = 2 }}\pi (19) = 38 \pi\)

Circumference of\(2^\rm{ nd }\) circle \(= 2 \pi{{\text{r}}_{\text{2}}}{\text{ = 2 }}\pi (9) = 18 \pi\)

Circumference of \(3^\rm{rd}\)circle \(=2\pi r\)

Given that,

Circumference of \(3^\rm{rd}\) \(\rm{}circle\) \(=\) Circumference of \(1^\rm{st }\) \(\rm{}circle\) \(+\)Circumference of \(2^\text{nd}\)\(\rm{}circle.\)

\[\begin{align} 2 \pi r &=38 \pi+18 \pi \\ &=56 \pi \\ r &=\frac{56 \pi}{2 \pi} \\ &=28 \end{align}\]

Therefore, the radius of the circle which has circumference equal to the sum of the circumference of the given two circles is \(28\, \rm{cm.}\)