A point P is 25 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 24 cm. Find the radius of the circle.

Options

A. 5 cm
B. 7 cm
C. 6 cm
D. 8 cm

Answer

To find the radius of the circle given that a point P is 25 cm from the center of the circle and the length of the tangent drawn from P to the circle is 24 cm, we can use the Pythagorean theorem. Here’s how to solve it step by step:

### Step 1: Understand the Components
- Let O be the center of the circle.
- Let A be the point where the tangent from P touches the circle.
- OP is the distance from the center O to point P, which is given as 25 cm.
- PA is the length of the tangent from P to the circle, which is given as 24 cm.
- OA is the radius of the circle, which we need to find and denote as R.

### Step 2: Formulate the Right Triangle
In triangle OAP:
- OP is the hypotenuse.
- OA is one leg (the radius).
- PA is the other leg (the tangent).

Since the tangent is perpendicular to the radius at the point of contact, we have a right triangle.

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ OP^2 = OA^2 + PA^2 \]

Substituting the known values:
\[ 25^2 = R^2 + 24^2 \]

### Step 4: Calculate the Squares
Calculating the squares:
- \( 25^2 = 625 \)
- \( 24^2 = 576 \)

### Step 5: Substitute and Rearrange
Now substitute these values into the equation:
\[ 625 = R^2 + 576 \]

Rearranging gives:
\[ R^2 = 625 - 576 \]

### Step 6: Perform the Subtraction
Calculating the right side:
\[ R^2 = 49 \]

### Step 7: Find the Radius
Taking the square root of both sides:
\[ R = \sqrt{49} \]
\[ R = 7 \]

### Conclusion
The radius of the circle is 7 cm.

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