Two circles of radii 4 cm and 9 cm respectively touch each other externally at a point and a common tangent touches them at a point P and Q respectively Then, area of square with one side PQ is

Options

A. 72 sq cm
B. 144 sq cm
C. 97 sq cm
D. 194 sq cm

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step-by-Step Solution:

1. **Identify the Radii of the Circles**:
- The radius of the first circle (r1) = 4 cm
- The radius of the second circle (r2) = 9 cm

2. **Determine the Distance Between the Centers of the Circles**:
- Since the circles touch each other externally, the distance between their centers (AB) is the sum of their radii:
\[ AB = r1 + r2 = 4 \, \text{cm} + 9 \, \text{cm} = 13 \, \text{cm} \]

3. **Identify Points on the Tangent**:
- Let P be the point where the tangent touches the first circle and Q be the point where it touches the second circle.
- Draw a perpendicular line from the center of the first circle (A) to the tangent at point P, and let this intersection point be L.
- Similarly, draw a perpendicular line from the center of the second circle (B) to the tangent at point Q.

4. **Determine the Length of AL**:
- Since AL is the radius of the first circle, we have:
\[ AL = r1 = 4 \, \text{cm} \]

5. **Determine the Length of BL**:
- Since BL is the radius of the second circle, we have:
\[ BL = r2 = 9 \, \text{cm} \]

6. **Calculate the Length of AL**:
- The distance AL can be calculated as:
\[ AL = r2 - r1 = 9 \, \text{cm} - 4 \, \text{cm} = 5 \, \text{cm} \]

7. **Apply the Pythagorean Theorem**:
- In triangle ABL, we can apply the Pythagorean theorem to find the length of BL:
\[ AB^2 = AL^2 + BL^2 \]
- Rearranging gives:
\[ BL^2 = AB^2 - AL^2 \]
- Substituting the known values:
\[ BL^2 = 13^2 - 5^2 = 169 - 25 = 144 \]
- Therefore,
\[ BL = \sqrt{144} = 12 \, \text{cm} \]

8. **Determine the Length of PQ**:
- Since PQ is equal to BL (as both are perpendicular to the radius at points P and Q), we have:
\[ PQ = BL = 12 \, \text{cm} \]

9. **Calculate the Area of the Square**:
- The area of the square with side PQ is given by:
\[ \text{Area} = PQ^2 = 12^2 = 144 \, \text{cm}^2 \]

### Final Answer:
The area of the square with one side PQ is **144 cm²**.