Finding the Radius of a Circle Given Chord Length and Distance from Center

This problem involves understanding the relationship between the radius of a circle, a chord within the circle, and the perpendicular distance from the center of the circle to that chord.

In a circle, a chord is a line segment connecting two points on the circumference. The distance from the center to the chord is the length of the perpendicular segment from the center to the chord.

A key property in circle geometry is that the perpendicular from the center of a circle to a chord bisects the chord. This means it cuts the chord into two equal halves.

Let's break down the given information:

• Length of the chord = 14 cm
• Distance from the center to the chord = 24 cm

Since the perpendicular from the center bisects the chord, half the length of the chord will be:

\( \text{Half chord length} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \)

Now, consider the right-angled triangle formed by:

• The radius of the circle (hypotenuse)
• Half the length of the chord (one leg)
• The distance from the center to the chord (the other leg)

Let \( r \) be the radius of the circle. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Applying the Pythagorean theorem:

\( r^2 = (\text{Half chord length})^2 + (\text{Distance from center})^2 \)

\( r^2 = (7 \text{ cm})^2 + (24 \text{ cm})^2 \)

\( r^2 = 49 \text{ cm}^2 + 576 \text{ cm}^2 \)

\( r^2 = 625 \text{ cm}^2 \)

To find the radius \( r \), we take the square root of 625:

\( r = \sqrt{625 \text{ cm}^2} \)

\( r = 25 \text{ cm} \)

Therefore, the length of the radius of the circle is 25 cm.

Steps to Calculate Circle Radius

1. Identify the given chord length and the distance from the circle's center to the chord.
2. Calculate half of the chord length, as the perpendicular from the center bisects the chord.
3. Recognize that the radius, half-chord, and distance from the center form a right-angled triangle with the radius as the hypotenuse.
4. Apply the Pythagorean theorem (\( a^2 + b^2 = c^2 \)), where \( a \) is half the chord length, \( b \) is the distance from the center, and \( c \) is the radius.
5. Solve the equation for the radius.

Revision Table: Circle Geometry Facts

Additional Information: Pythagorean Triples

A Pythagorean triple is a set of three positive integers \( a \), \( b \), and \( c \), such that \( a^2 + b^2 = c^2 \). These triples represent the side lengths of a right-angled triangle. Common Pythagorean triples can help quickly identify solutions in geometry problems involving right triangles without lengthy calculations.

Examples of Pythagorean triples include:

• (3, 4, 5)
• (5, 12, 13)
• (7, 24, 25)
• (8, 15, 17)

In this problem, we had legs of length 7 cm and 24 cm. Recognizing that (7, 24, 25) is a Pythagorean triple allows us to quickly see that the hypotenuse (the radius) must be 25 cm.