Understanding the Geometry of a Circle, Chord, and Radius

This problem involves finding the radius of a circle given the length of a chord and the perpendicular distance from the circle's center to that chord. This setup creates a right-angled triangle, allowing us to use the Pythagorean theorem.

Relationship Between Chord, Distance, and Radius

Consider a circle with center O. Let AB be a chord of the circle. If we draw a line segment from the center O perpendicular to the chord AB, let the point of intersection be M. A fundamental property of circles states that the perpendicular from the center to a chord bisects the chord. This means that M is the midpoint of AB, so AM = MB = (1/2) * AB.

In this scenario, we have:

• OA (or OB) is the radius of the circle, which we want to find (let's call it 'r').
• OM is the perpendicular distance from the center to the chord (given as 5 cm).
• AM (or MB) is half the length of the chord (half of 24 cm).

The line segments OM, AM, and OA form a right-angled triangle ▵OMA, with the right angle at M (since OM is perpendicular to AB).

Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In ▵OMA, OA is the hypotenuse, OM is one leg, and AM is the other leg.

So, according to the Pythagorean theorem:

\( OA^2 = OM^2 + AM^2 \)

Substituting the variables:

\( r^2 = d^2 + (\frac{c}{2})^2 \)

Where:

• \( r \) is the radius (unknown)
• \( d \) is the perpendicular distance from the center to the chord (given as 5 cm)
• \( c \) is the length of the chord (given as 24 cm)

Step-by-Step Calculation of the Radius

Let's plug in the given values:

Chord length \( c = 24 \) cm

Perpendicular distance \( d = 5 \) cm

Half the chord length \( \frac{c}{2} = \frac{24}{2} = 12 \) cm

Now, substitute these values into the Pythagorean theorem formula:

\( r^2 = d^2 + (\frac{c}{2})^2 \)

\( r^2 = 5^2 + 12^2 \)

Calculate the squares:

\( 5^2 = 5 \times 5 = 25 \)

\( 12^2 = 12 \times 12 = 144 \)

Add the squared values:

\( r^2 = 25 + 144 \)

\( r^2 = 169 \)

To find \( r \), take the square root of both sides:

\( r = \sqrt{169} \)

The square root of 169 is 13.

\( r = 13 \)

So, the radius of the circle is 13 cm.

Therefore, the radius of the circle is 13 cm.

Revision Table: Key Concepts

Additional Information: Circle Geometry and Theorems

Understanding basic circle geometry is crucial for solving problems like this. Here are some related points:

• The longest chord in a circle is the diameter, which passes through the center.
• The perpendicular bisector of a chord always passes through the center of the circle.
• If two chords are equidistant from the center, then they are equal in length. Conversely, if two chords are equal in length, they are equidistant from the center.
• Many circle theorems relate angles, arcs, chords, and tangents. The relationship between the chord, its distance from the center, and the radius is a fundamental application of the Pythagorean theorem in circle geometry.

Practicing different problems involving chords, radii, and tangents will help solidify your understanding of these concepts.