Question

From point Q which is outside the circle, the length of the tangent to the circle is 24 cm and the distance of Q from the centre is 25 cm, then the radius of the circle is given by

• A. 7 cm
• B. 12 cm
• C. 15 cm
• D. 24.5 cm

Hint:

This problem utilizes the famous Pythagorean triple:
\[ (7, 24, 25) \] Knowing common Pythagorean triples (like \( (3, 4, 5) \), \( (5, 12, 13) \), and \( (7, 24, 25) \)) allows you to write down the answer instantly without performing any calculations.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires applying the properties of circle geometry, specifically the relationship between a circle's radius and a tangent line drawn from an external point.
Key Formula or Approach:
A fundamental property of circles states that a tangent line is always perpendicular to the radius at the point of contact.
This forms a right-angled triangle where:
- One leg is the radius of the circle (\( r \)).
- The other leg is the length of the tangent line (\( t \)).
- The hypotenuse is the distance from the external point to the center of the circle (\( d \)).
We can solve for the missing side using the Pythagorean Theorem:
\[ r^2 + t^2 = d^2 \]

Step 2: Detailed Explanation:
Let us identify the given values:
- Length of the tangent line, \( t = 24\text{ cm} \)
- Distance from point \( Q \) to the center of the circle, \( d = 25\text{ cm} \)
Let \( r \) be the radius of the circle. Apply the Pythagorean theorem to this right-angled triangle:
\[ r^2 + t^2 = d^2 \] Substitute the given values into the equation:
\[ r^2 + 24^2 = 25^2 \] Calculate the squares:
\[ 24^2 = 576 \] \[ 25^2 = 625 \] Substitute these values back into the equation:
\[ r^2 + 576 = 625 \] Isolate \( r^2 \):
\[ r^2 = 625 - 576 \] \[ r^2 = 49 \] Take the square root of both sides (since radius must be a positive length):
\[ r = \sqrt{49} = 7\text{ cm} \]

Step 3: Final Answer:
The radius of the circle is \( 7\text{ cm} \).
Therefore, the correct choice is Option (A).