The distance of point A from the centre of the circle is 5 cm. The length of the tangent is 4 cm. The radius of the circle is :

Options

A. 3 cm
B. 4 cm
C. 5 cm
D. 8 cm

Answer

To find the radius of the circle given the distance from point A to the center O (5 cm) and the length of the tangent from point A to the circle (4 cm), we can use the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the components:**
- Let O be the center of the circle.
- Let A be the point outside the circle.
- Let C be the point where the tangent touches the circle.
- We know:
- Distance AO = 5 cm (distance from point A to the center O)
- Length of tangent AC = 4 cm (length of the tangent from A to C)
- Let the radius of the circle be R (OC).

2. **Apply the Pythagorean theorem:**
- According to the Pythagorean theorem in triangle AOC:
\[ AO^2 = AC^2 + OC^2 \]
- Substituting the known values:
\[ 5^2 = 4^2 + R^2 \]

3. **Calculate the squares:**
- Calculate \(5^2\) and \(4^2\):
\[ 25 = 16 + R^2 \]

4. **Rearrange the equation to solve for R^2:**
- Subtract 16 from both sides:
\[ R^2 = 25 - 16 \]
\[ R^2 = 9 \]

5. **Find R by taking the square root:**
- Taking the square root of both sides:
\[ R = \sqrt{9} \]
\[ R = 3 \text{ cm} \]

### Final Answer:
The radius of the circle is 3 cm.

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