Understanding the Problem: Distance of a Chord from Circle Centre

The question asks us to find the distance between the centre of a circle and a chord within it, given the circle's radius and the chord's length. We are provided with a circle having a radius of 5 cm and a chord AB with a length of 6 cm.

Key Geometric Principle: Perpendicular from the Centre

A fundamental property of circles is that a line segment drawn from the centre perpendicular to a chord bisects the chord. This means it divides the chord into two equal parts.

Let's visualize this:

• Let O be the centre of the circle.
• Let AB be the chord.
• Draw a line segment from O perpendicular to AB. Let this segment meet AB at point M.
• Since OM is perpendicular to AB, M is the midpoint of AB.

Because M is the midpoint of the chord AB (which is 6 cm long), the length of AM will be half the length of AB.

Length of AM $= \frac{\text{Length of AB}}{2} = \frac{6 \text{ cm}}{2} = 3 \text{ cm}$.

Forming a Right-Angled Triangle

The line segments OA (which is the radius), AM (half the chord length), and OM (the distance we need to find) form a right-angled triangle, ▵OMA, at point M. The radius OA is the hypotenuse of this right-angled triangle.

• Hypotenuse (OA) = Radius = 5 cm
• One leg (AM) = Half of chord length = 3 cm
• Other leg (OM) = Distance of the chord from the centre = Let's call it 'd'.

Applying 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. This is known as the Pythagorean Theorem. For ▵OMA, it can be written as:

$\text{OA}^2 = \text{AM}^2 + \text{OM}^2$

Substituting the known values:

$5^2 = 3^2 + d^2$

Calculating the Distance

Now, we solve the equation for 'd':

$25 = 9 + d^2$

$d^2 = 25 - 9$

$d^2 = 16$

To find 'd', we take the square root of both sides:

$d = \sqrt{16}$

$d = 4 \text{ cm}$

Therefore, the distance of the chord AB from the centre of the circle is 4 cm.

Summary of Steps for Chord Distance Calculation

1. Identify the radius and the length of the chord.
2. Calculate half the length of the chord.
3. Recognize that the radius, half-chord, and distance from the centre form a right-angled triangle.
4. Apply the Pythagorean theorem using the radius as the hypotenuse and half-chord as one leg.
5. Solve for the unknown distance.

Revision Table: Circle Geometry Basics

Additional Information: Chord Properties and Applications

Understanding the relationship between the circle's center, radius, and chords is crucial in geometry. Here are some related points:

• Longest Chord: The diameter is the longest chord in a circle. It passes through the center and its length is twice the radius.
• Congruent Chords: Chords that are equidistant from the center are congruent (have the same length). Conversely, congruent chords are equidistant from the center.
• Perpendicular Bisector: The perpendicular bisector of any chord always passes through the center of the circle.
• Calculating Chord Length: If you know the radius (r) and the distance (d) of a chord from the center, you can find the chord length (L) using the formula: $L = 2\sqrt{r^2 - d^2}$. This is derived directly from the Pythagorean theorem.

These principles are widely used in solving various problems involving circles and their parts.