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.

Let us given that, 

Radius of the first circle = 19 cm

Radius of the second circle = 9 cm

We know that, Circumference of a circle = 2$\mathrm{\pi }$r 

Consider,

The radius of the third circle is r.

Circumference of 1^st circle = 2$\mathrm{\pi }$r = 38 $\mathrm{\pi }$

Circumference of 2^nd circle = 2$\mathrm{\pi }$r = 18 $\mathrm{\pi }$

Circumference of 3^rd circle = 2$\mathrm{\pi }$r 

Now, it is given that

Circumference of 3^rd circle = Circumference of 1^st circle  $+$ Circumference of 2^nd circle

2$\mathrm{\pi }$r  =  38 $\mathrm{\pi }$ $+$ 18 $\mathrm{\pi }$

2$\mathrm{\pi }$r  = 56 $\mathrm{\pi }$

r = 56$\mathrm{\pi }$/2$\mathrm{\pi }$

r = 28 cm

Hence, the radius of the circle that has a circumference equal to the sum of the circumference of the two given circles is 28 cm.