PT is a tangent to a circle with centre O and radius 6cm. If PT is 8cm then length of OP is:

Options

A. a.10cm
B. b. 12cm
C. c. 16 cm
D. d.none of these

Answer

To solve the problem step by step, we will use the properties of tangents and the Pythagorean theorem.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a circle with center O and radius 6 cm. PT is a tangent to the circle, and its length is 8 cm. We need to find the length of OP, where P is the point outside the circle.

2. **Identify Key Points**:
- O is the center of the circle.
- T is the point where the tangent PT touches the circle.
- The radius OT is perpendicular to the tangent PT at point T.

3. **Draw a Right Triangle**:
- We can visualize the situation as a right triangle OTP, where:
- OT is the radius (6 cm),
- PT is the tangent (8 cm),
- OP is the hypotenuse (which we need to find).

4. **Apply the Pythagorean Theorem**:
- According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (OP) is equal to the sum of the squares of the lengths of the other two sides (OT and PT).
- The formula is:
\[ OP^2 = OT^2 + PT^2 \]

5. **Substitute the Known Values**:
- We know OT = 6 cm and PT = 8 cm.
- Substitute these values into the equation:
\[ OP^2 = 6^2 + 8^2 \]

6. **Calculate the Squares**:
- Calculate \(6^2\) and \(8^2\):
\[ 6^2 = 36 \]
\[ 8^2 = 64 \]

7. **Add the Squares**:
- Now, add the results:
\[ OP^2 = 36 + 64 = 100 \]

8. **Find the Length of OP**:
- To find OP, take the square root of both sides:
\[ OP = \sqrt{100} = 10 \text{ cm} \]

9. **Conclusion**:
- The length of OP is 10 cm.

### Final Answer:
The length of OP is **10 cm**.

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