From a point Q, the length of tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. What is the radius of the circle?

Steps to Solve

1. Visualize the problem Imagine a circle with center O. From point Q outside the circle, a tangent line is drawn to the circle, touching the circle at point P. We are given the length of the tangent QP = 24 cm and the distance from Q to the center O, which is OQ = 25 cm. We need to find the radius of the circle, OP.

2. Recognize the right triangle Since QP is tangent to the circle at P, the radius OP is perpendicular to the tangent line QP. Therefore, triangle OPQ is a right-angled triangle with the right angle at P.

3. Apply the Pythagorean theorem In right triangle OPQ, we have $OP^2 + PQ^2 = OQ^2$, where OP is the radius we want to find. We are given PQ = 24 cm and OQ = 25 cm. Let OP = r.

4. Substitute the given values into the Pythagorean theorem Substitute PQ = 24 and OQ = 25 into the equation: $r^2 + 24^2 = 25^2$

5. Solve for $r^2$ Calculate $24^2$ and $25^2$: $24^2 = 576$ $25^2 = 625$ So, the equation becomes: $r^2 + 576 = 625$

6. Isolate $r^2$ Subtract 576 from both sides of the equation: $r^2 = 625 - 576$ $r^2 = 49$

7. Solve for r Take the square root of both sides of the equation: $r = \sqrt{49}$ $r = 7$

Therefore, the radius of the circle is 7 cm.