The length of the tangent to a circle from a point P, which is 25 cm away from the centre, is 24 cm. What is 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 (AP) = 24 cm
- Distance from point P to the center O (OP) = 25 cm

2. **Draw the Diagram**:
- Draw a circle with center O.
- Mark point P outside the circle such that OP = 25 cm.
- Draw the tangent line from point P to the circle, touching the circle at point A. The length of the tangent PA = 24 cm.

3. **Apply the Right Triangle Concept**:
- The radius OA is perpendicular to the tangent PA at point A. Therefore, triangle OAP is a right triangle with:
- OP as the hypotenuse,
- OA as one leg (the radius we need to find),
- AP as the other leg (the length of the tangent).

4. **Use the Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ OP^2 = OA^2 + AP^2 \]
- Substitute the known values:
\[ 25^2 = OA^2 + 24^2 \]

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

6. **Rearrange the Equation**:
- To find \(OA^2\):
\[ OA^2 = 625 - 576 \]
\[ OA^2 = 49 \]

7. **Find the Radius**:
- Take the square root of both sides to find OA (the radius):
\[ OA = \sqrt{49} = 7 \text{ cm} \]

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

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