Question

The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Answer 1

Hint:The circumference of a circle is given by $2\pi r$, where r is the radius of the circle. If circumference of a circle is equal to the sum of circumference of any two circles, then following formula is used to calculate the radius of the circle: $2\pi r = 2\pi {r_1} + 2\pi {r_2}$, where $r_1$ and $r_2$ are the radius of two circles.

Complete step-by-step answer:
Given, Radius of 1st circle \[{r_1} = 19\]cm
Radius of 2nd circle \[{r_2} = 9\,\]cm
Circumference of 1st circle= $2\pi {r_1}$
$ \Rightarrow $ Circumference of 1st circle= $2\pi \times 19$
$ \Rightarrow $ Circumference of 1st circle= $38\pi $

Similarly, circumference of 2nd circle=$2\pi {r_2}$
$ \Rightarrow $Circumference of 2nd circle=$2\pi \times 9$
$ \Rightarrow $Circumference of 2nd circle=$18\pi $
Let the radius of required circle= $r$cm
$\therefore $ Circumference of the required circle= $2\pi r$
According to the question,
 Circumference of the required circle= Sum of the circumference of the two circles
$ \Rightarrow $$2\pi r = 2\pi {r_1} + 2\pi {r_2}$
$ \Rightarrow 2\pi r = 38\pi + 18\pi $
$ \Rightarrow $$2\pi r = 56\pi $
$ \Rightarrow $$r = \dfrac{{56\pi }}{{2\pi }}$
$ \Rightarrow $$r = 28$cm
Hence, the radius of the required circle is $r = 28$cm.

Note:Circumference of a circle is the measurement of the boundary of the circle. If we open a circle and make a straight line out of it, then its length is the circumference.Students should remember the definitions and formulas related to the circle for solving these types of questions.