Question

A tangent \( PQ \) at a point \( P \) of a circle of radius 9 cm meets a line through the center \( O \) at a point \( Q \) such that \( OQ = 15 \) cm. The length of \( PQ \) is:

• A. \( 12 \, \text{cm} \)
• B. \( 13 \, \text{cm} \)
• C. \( 24 \, \text{cm} \)
• D. \( 25 \, \text{cm} \)

Hint:

For tangents to a circle, use the Pythagorean theorem to relate the radius, the tangent, and the distance from the center.

Answer 1

The Correct Option is A

Step 1: Understand the problem setup
We are given a circle with a radius of 9 cm, and a tangent \( PQ \) at a point \( P \) on the circle.
The line \( OQ \) passes through the center \( O \) of the circle, and \( OQ = 15 \, \text{cm} \).
We need to determine the length of the tangent \( PQ \).
Step 2: Apply the Pythagorean theorem
Since \( PQ \) is tangent to the circle at point \( P \), and \( OQ \) passes through the center, we form a right triangle \( OPQ \).
In this triangle, \( OP \) is the radius of the circle, \( PQ \) is the tangent, and \( OQ \) is the hypotenuse.
\[ OQ^2 = OP^2 + PQ^2 \] Substitute the known values:
\( OQ = 15 \, \text{cm} \),
\( OP = 9 \, \text{cm} \) (radius of the circle).
\[ 15^2 = 9^2 + PQ^2 \] \[ 225 = 81 + PQ^2 \] \[ PQ^2 = 225 - 81 = 144 \] \[ PQ = \sqrt{144} = 12 \, \text{cm} \] Thus, the length of \( PQ \) is \( 12 \, \text{cm} \).