Step-by-step Solution:

To find the radius of a new circle whose circumference is equal to the sum of the circumferences of two given circles,
we can use the formula for the circumference of a circle:
\[ \text{Circumference} = 2\pi r \]
Given:
Radius of first circle \( r_1 = 19 \) cm
Radius of second circle \( r_2 = 9 \) cm
1. Circumference of first circle: \[ C_1 = 2\pi \cdot 19 = 38\pi \] 2. Circumference of second circle: \[ C_2 = 2\pi \cdot 9 = 18\pi \] 3. Total circumference: \[ C = C_1 + C_2 = 38\pi + 18\pi = 56\pi \] 4. Let the radius of the new circle be \( R \). Then, \[ 2\pi R = 56\pi \] Divide both sides by \( 2\pi \): \[ R = \frac{56\pi}{2\pi} = 28 \]