Question

A tangent AB at a point 'A' of a circle of radius 7 cm meets a line through the centre 'C' at a point 'B' so that CB=11 cm, then the length of AB =

• A. \(\sqrt{71}\) cm
• B. \(6\sqrt{2}\) cm
• C. 9 cm
• D. 12 cm

Answer 1

The Correct Option is B

To solve the problem, we need to find the length of the tangent \( AB \) from the point of contact \( A \) on the circle to a point \( B \) on a line through the center \( C \), where \( CB = 11 \, \text{cm} \) and the radius \( CA = 7 \, \text{cm} \).

1. Understanding the Geometry:
The line segment \( AB \) is a tangent to the circle at point \( A \). The radius \( CA \) is perpendicular to the tangent at the point of contact \( A \). Therefore, triangle \( \triangle ABC \) is a right-angled triangle at \( A \).

2. Applying the Pythagorean Theorem:
In right-angled triangle \( \triangle ABC \), we use the Pythagorean theorem:

\( CB^2 = CA^2 + AB^2 \)

3. Substituting Known Values:
Given \( CB = 11 \, \text{cm} \), and \( CA = 7 \, \text{cm} \), substitute into the equation:

\( 11^2 = 7^2 + AB^2 \)
\( 121 = 49 + AB^2 \)

4. Solving for \( AB \):
\( AB^2 = 121 - 49 = 72 \)
\( AB = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \, \text{cm} \)

Final Answer:
The length of \( AB \) is \({6\sqrt{2} \, \text{cm}} \).