From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle.

Answer

To find the radius of the circle given the length of the tangent and the distance from the point to the center, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of the tangent (PQ) = 24 cm
- Distance from point Q to the center O (OQ) = 25 cm

2. **Understand the Geometry:**
- The tangent at point P is perpendicular to the radius OP at point P. Therefore, triangle OPQ is a right triangle with:
- OP as the radius (R)
- PQ as the length of the tangent
- OQ as the distance from the center to point Q

3. **Apply the Pythagorean Theorem:**
- According to the Pythagorean theorem, in triangle OPQ:
\[ OQ^2 = OP^2 + PQ^2 \]
- Substituting the known values:
\[ 25^2 = R^2 + 24^2 \]

4. **Calculate the Squares:**
- Calculate \(25^2\) and \(24^2\):
\[ 25^2 = 625 \]
\[ 24^2 = 576 \]

5. **Set Up the Equation:**
- Substitute the squares into the equation:
\[ 625 = R^2 + 576 \]

6. **Isolate R²:**
- Rearranging the equation to solve for \(R^2\):
\[ R^2 = 625 - 576 \]
\[ R^2 = 49 \]

7. **Find the Radius R:**
- Take the square root of both sides:
\[ R = \sqrt{49} = 7 \text{ cm} \]

### Final Answer:
The radius of the circle is **7 cm**.