The radii of two circles are 5 cm and 12 cm. The area of a third circle is equal to the sum of the areas 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 need to follow these steps:

### Step 1: Identify the radii of the two circles.
- The radius of the first circle (r1) is given as 5 cm.
- The radius of the second circle (r2) is given as 12 cm.

### Step 2: Calculate the areas of the two circles.
- The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
- For the first circle:
\[ \text{Area}_1 = \pi (5)^2 = \pi \times 25 = 25\pi \text{ cm}^2 \]
- For the second circle:
\[ \text{Area}_2 = \pi (12)^2 = \pi \times 144 = 144\pi \text{ cm}^2 \]

### Step 3: Find the total area of the two circles.
- The total area of the two circles is:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 25\pi + 144\pi = 169\pi \text{ cm}^2 \]

### Step 4: Set the area of the third circle equal to the total area of the first two circles.
- Let the radius of the third circle be \( r3 \). Then, the area of the third circle is:
\[ \text{Area}_3 = \pi (r3)^2 \]
- According to the question:
\[ \pi (r3)^2 = 169\pi \]

### Step 5: Simplify the equation.
- We can divide both sides of the equation by \( \pi \):
\[ (r3)^2 = 169 \]

### Step 6: Solve for \( r3 \).
- Taking the square root of both sides:
\[ r3 = \sqrt{169} = 13 \text{ cm} \]

### Conclusion:
- The radius of the third circle is 13 cm.

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