Question

The length of the tangent from a point 15 cm away from the centre of a circle of radius 9 cm is:

• A. 15 cm
• B. 13 cm
• C. 12 cm
• D. 11 cm

Hint:

To find the length of a tangent from a point outside the circle, use the formula \(\sqrt{d^2 - r^2}\), where \(d\) is the distance from the point to the center and \(r\) is the radius of the circle.

Answer 1

The Correct Option is C

Step 1: Visualize the Problem Setup
Consider a circle with center O and radius r = 9 cm. A point P is 15 cm away from O. A tangent from P touches the circle at point T. Since a tangent is perpendicular to the radius at the point of contact, angle OTP = 90 degrees, forming a right triangle OTP. In this triangle:
- OT = 9 cm (radius),
- OP = 15 cm (distance from the point to the center),
- PT is the length of the tangent, which we need to find.

Step 2: Use the Pythagorean Theorem
In right triangle OTP, apply the Pythagorean theorem:
OP² = OT² + PT²
Rearrange to solve for PT:
PT² = OP² - OT²
Substitute the values:
- OP = 15 cm, so OP² = 15² = 225,
- OT = 9 cm, so OT² = 9² = 81.
Then:
PT² = 225 - 81 = 144
PT = √144 = 12 cm

Step 3: Interpret the Result
The length of the tangent is 12 cm. Since OP = 15 cm is greater than the radius 9 cm, point P is outside the circle, which is consistent with the problem. The calculated length of 12 cm matches option 3 from the given choices: 15 cm, 13 cm, 11 cm, and 12 cm.

Final Answer: The length of the tangent is 12 cm (Option 3).