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Hint: Assume a variable r that will represent the radius of the circle whose circumference is equal to the sum of the circumference of the circles having radius 19 cm and 9 cm. For a circle with radius equal to r, the circumference is given by the formula $2\pi r$. Using this formula, the question can be solved.

__Complete step-by-step answer:__

Before proceeding with the question, we must know all the formulas that will be required to solve this question.

Consider a circle having its radius equal to r. To find the circumference of this circle, we use the formula,

C = $2\pi r$ . . . . . . . . . . . (1)

In the question, we have to find the radius of the circle whose circumference is equal to the sum of the circumference of the circles having radius 19 cm and 9 cm.

Let us assume the radius of this circle as r. Using formula (1), the circumference of this circle is,

C = $2\pi r$ . . . . . . . . (2)

Using formula (1), the circumference of the circle with radius 19 cm is equal to,

${{C}_{1}}=2\pi \left( 19 \right)$ . . . . . . . . (3)

Using formula (1), the circumference of the circle with radius 9 cm is equal to,

${{C}_{2}}=2\pi \left( 9 \right)$ . . . . . . . . (4)

In the question, it is given that,

$C={{C}_{1}}+{{C}_{2}}$

Substituting C from equation (2), ${{C}_{1}}$ from equation (3) and ${{C}_{2}}$ from equation (4) in the above equation, we get,

$\begin{align}

& 2\pi r=2\pi \left( 19 \right)+2\pi \left( 9 \right) \\

& \Rightarrow 2\pi r=2\pi \left( 19+9 \right) \\

& \Rightarrow 2\pi r=2\pi \left( 28 \right) \\

& \Rightarrow r=28cm \\

\end{align}$

Hence, the answer is 28 cm.

Note: This question can be done directly if one knows that the circumference is linearly proportional to the radius of the circle. Since the circumference of the first circle is equal to the sum of the circumference of the other two circles, we can say that the radius of the first circle is equal to the sum of the radius of the other two circles.

Before proceeding with the question, we must know all the formulas that will be required to solve this question.

Consider a circle having its radius equal to r. To find the circumference of this circle, we use the formula,

C = $2\pi r$ . . . . . . . . . . . (1)

In the question, we have to find the radius of the circle whose circumference is equal to the sum of the circumference of the circles having radius 19 cm and 9 cm.

Let us assume the radius of this circle as r. Using formula (1), the circumference of this circle is,

C = $2\pi r$ . . . . . . . . (2)

Using formula (1), the circumference of the circle with radius 19 cm is equal to,

${{C}_{1}}=2\pi \left( 19 \right)$ . . . . . . . . (3)

Using formula (1), the circumference of the circle with radius 9 cm is equal to,

${{C}_{2}}=2\pi \left( 9 \right)$ . . . . . . . . (4)

In the question, it is given that,

$C={{C}_{1}}+{{C}_{2}}$

Substituting C from equation (2), ${{C}_{1}}$ from equation (3) and ${{C}_{2}}$ from equation (4) in the above equation, we get,

$\begin{align}

& 2\pi r=2\pi \left( 19 \right)+2\pi \left( 9 \right) \\

& \Rightarrow 2\pi r=2\pi \left( 19+9 \right) \\

& \Rightarrow 2\pi r=2\pi \left( 28 \right) \\

& \Rightarrow r=28cm \\

\end{align}$

Hence, the answer is 28 cm.

Note: This question can be done directly if one knows that the circumference is linearly proportional to the radius of the circle. Since the circumference of the first circle is equal to the sum of the circumference of the other two circles, we can say that the radius of the first circle is equal to the sum of the radius of the other two circles.

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