Question

In the given figure, PA is the tangent drawn from an external point P to the circle with center O. If the radius of the circle is 3 cm and PA = 4 cm, then the length of PB is

• A. 3 cm
• B. 2 cm
• C. 5 cm
• D. 4 cm

Answer 1

The Correct Option is D

Given: PA is the tangent from an external point \( P \) to a circle with center \( O \). The radius of the circle is 3 cm, and the length of \( PA = 4 \) cm.

Step 1: Understanding the right-angled triangle 

Since \( PA \) is a tangent to the circle at point \( A \), the radius \( OA \) is perpendicular to the tangent \( PA \). This forms a right-angled triangle \( OPA \), where:

• \( OP \) is the hypotenuse
• \( OA = 3 \) cm (radius)
• \( PA = 4 \) cm (tangent length)

Step 2: Applying the Pythagoras Theorem

\[ OP^2 = OA^2 + PA^2 \] \[ OP^2 = 3^2 + 4^2 \] \[ OP^2 = 9 + 16 = 25 \] \[ OP = \sqrt{25} = 5 \text{ cm} \]

Step 3: Finding PB

Since two tangents drawn from an external point to a circle are equal in length, we have:

\[ PB = PA = 4 \text{ cm} \]

Final Answer: 4 cm