From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm?

Question

Answer

Answer

Answer

Given:
- Length of tangent to the circle from point Q = 24 cm
- Distance of point Q from the center of the circle = 25 cm

To find:
- Radius of the circle

Approach:
- Draw a diagram to visualize the given information.
- Use the properties of tangents and the Pythagorean theorem to solve the problem.
- Set up an equation using the given information and solve for the radius of the circle.

Solution:

Step 1: Draw a diagram
- Draw a circle with center O.
- Mark the point Q outside the circle.
- Draw a line segment from the center O to the point Q.
- Draw a tangent line from point Q to touch the circle at point T.
- Mark the length of the tangent line as 24 cm.
- Mark the distance of point Q from the center O as 25 cm.

Step 2: Use properties of tangents
- The tangent line from point Q to the circle is perpendicular to the radius drawn from the center O to the point of contact T.
- Therefore, triangle OQT is a right-angled triangle.

Step 3: Apply the Pythagorean theorem
- In right-angled triangle OQT, apply the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- In this case, the hypotenuse is the line segment QT, which is the tangent to the circle.
- The other two sides are the radius of the circle (r) and the distance from the center O to the point Q (25 cm).
- So, we have the equation: r^2 + 25^2 = 24^2

Step 4: Solve the equation
- Simplify the equation: r^2 + 625 = 576
- Subtract 625 from both sides: r^2 = 576 - 625
- Simplify further: r^2 = -49
- Since the radius cannot be negative, we discard this solution.

Step 5: Choose the correct answer
- Since the equation has no valid solution, none of the given answer options (A, B, C, D) are correct.

Conclusion:
- The radius of the circle cannot be determined based on the given information.

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