Circle Radius Calculation

We are given a circle with a chord of length 24 cm and the distance from the circle's center to the chord is 5 cm. We need to find the radius of the circle.

Geometric Properties

Key properties used:

• A perpendicular line from the center of a circle to a chord bisects the chord.
• The radius, the perpendicular distance from the center to the chord, and half the chord form a right-angled triangle.

Applying the Pythagorean Theorem

First, find half the length of the chord:

Half chord length = $\frac{24 \text{ cm}}{2} = 12 \text{ cm}$

Now, consider the right-angled triangle formed by:

• Hypotenuse: Radius ($r$)
• One leg: Distance from the center ($d = 5$ cm)
• Other leg: Half the chord length ($\frac{L}{2} = 12$ cm)

Using the Pythagorean theorem, $r^2 = d^2 + (\frac{L}{2})^2$:

$ r^2 = (5 \text{ cm})^2 + (12 \text{ cm})^2 $

$ r^2 = 25 \text{ cm}^2 + 144 \text{ cm}^2 $

$ r^2 = 169 \text{ cm}^2 $

To find the radius ($r$), take the square root:

$ r = \sqrt{169 \text{ cm}^2} $

$ r = 13 \text{ cm} $

Conclusion

The radius of the circle is 13 cm.