The radii of two circles are 5 cm and 12 cm. The area of a third circle is equal to the sum of the area of the two circles. The radius of the third circle is :

Options

A. 13 cm
B. 21 cm
C. 30 cm
D. 17 cm

Answer

To find the radius of the third circle, we will first calculate the areas of the two given circles and then use that information to find the radius of the third circle.

### Step 1: Calculate the area of the first circle
The formula for the area of a circle is given by:
\[ \text{Area} = \pi r^2 \]
For the first circle with a radius of 5 cm:
\[ \text{Area}_1 = \pi (5)^2 = \pi \times 25 = 25\pi \text{ cm}^2 \]

### Step 2: Calculate the area of the second circle
For the second circle with a radius of 12 cm:
\[ \text{Area}_2 = \pi (12)^2 = \pi \times 144 = 144\pi \text{ cm}^2 \]

### Step 3: Calculate the total area of the two circles
Now, we will add the areas of the two circles:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 25\pi + 144\pi = 169\pi \text{ cm}^2 \]

### Step 4: Set the area of the third circle equal to the total area
Let the radius of the third circle be \( r \). The area of the third circle can be expressed as:
\[ \text{Area}_3 = \pi r^2 \]
Setting the area of the third circle equal to the total area of the first two circles:
\[ \pi r^2 = 169\pi \]

### Step 5: Solve for the radius \( r \)
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r^2 = 169 \]
Taking the square root of both sides:
\[ r = \sqrt{169} = 13 \text{ cm} \]

### Conclusion
The radius of the third circle is \( 13 \) cm.

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