Understanding the Circle Geometry Problem

The problem asks us to find the length of the diameter of a circle given the length of a chord and its distance from the center of the circle. This is a standard problem involving the properties of circles and the Pythagorean theorem.

Analyzing the Given Information

We are given:

• Length of the chord = 10 cm
• Distance of the chord from the center = 12 cm

We need to find the diameter of the circle.

Applying Circle Properties

A key property of a circle is that a line drawn from the center perpendicular to a chord bisects the chord. In this case, the distance of the chord from the center (12 cm) is measured along a line segment that is perpendicular to the chord and passes through the center.

When the perpendicular from the center meets the chord, it divides the chord into two equal parts. The length of each half of the chord is:

Half chord length \( = \frac{\text{Length of chord}}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \)

Forming a Right-Angled Triangle

Consider the following points:

• The center of the circle.
• One endpoint of the chord.
• The midpoint of the chord (where the perpendicular from the center meets the chord).

These three points form a right-angled triangle. The sides of this triangle are:

• One leg is the distance from the center to the chord (12 cm).
• The other leg is half the length of the chord (5 cm).
• The hypotenuse is the radius of the circle (the line segment from the center to an endpoint of the chord).

Using the Pythagorean Theorem to Find the Radius

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let \(r\) be the radius of the circle. According to the Pythagorean theorem:

\( \text{radius}^2 = (\text{distance from center})^2 + (\text{half chord length})^2 \)

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

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

\( r^2 = 169 \text{ cm}^2 \)

To find the radius, we take the square root of 169:

\( r = \sqrt{169} \text{ cm} \)

\( r = 13 \text{ cm} \)

So, the radius of the circle is 13 cm.

Calculating the Diameter

The diameter of a circle is twice its radius.

\( \text{Diameter} = 2 \times \text{Radius} \)

\( \text{Diameter} = 2 \times 13 \text{ cm} \)

\( \text{Diameter} = 26 \text{ cm} \)

Therefore, the length of the diameter of the circle is 26 cm.

Step-by-Step Solution Summary

1. Identify the given information: chord length (10 cm) and distance from center (12 cm).
2. Recall that the perpendicular from the center bisects the chord. Calculate half the chord length: \( \frac{10}{2} = 5 \) cm.
3. Recognize that the radius, the distance from the center, and half the chord form a right-angled triangle.
4. Apply the Pythagorean theorem \( r^2 = a^2 + b^2 \), where \(a\) is the distance from the center and \(b\) is half the chord length. \( r^2 = 12^2 + 5^2 = 144 + 25 = 169 \).
5. Calculate the radius \( r = \sqrt{169} = 13 \) cm.
6. Calculate the diameter: \( \text{Diameter} = 2 \times \text{Radius} = 2 \times 13 = 26 \) cm.

Revision Table: Circle Geometry Basics

Additional Information: Related Circle Concepts

Understanding the relationship between the radius, chord, and distance from the center is fundamental in circle geometry. Here are some related concepts:

• Perpendicular Bisector of a Chord: The perpendicular bisector of any chord of a circle always passes through the center of the circle.
• Congruent Chords: Chords that are equidistant from the center are congruent (have the same length). Conversely, congruent chords are equidistant from the center.
• Longest Chord: The longest chord in a circle is its diameter. It is the chord that passes through the center.
• Pythagorean Triples: In this problem, the sides of the right triangle are 5, 12, and 13. (5, 12, 13) is a common Pythagorean triple, meaning \(5^2 + 12^2 = 13^2\). Recognizing common triples can sometimes speed up calculations.

These properties are essential for solving various problems involving chords and distances within a circle.