A chord of circle of radius 7cm is 5cm from the centre of the circle.What is the length of the chord?

Analysis

To find the length of the chord, we need to visualize the circle. We are given:

• The radius of the circle ($r$) = 7 cm.
• The perpendicular distance from the center to the chord ($d$) = 5 cm.

Concept

When a line is drawn from the center of a circle perpendicular to a chord, it bisects the chord (divides it into two equal parts). This creates a right-angled triangle where:

• The hypotenuse is the radius of the circle.
• One leg is the distance from the center to the chord.
• The other leg is half the length of the chord ($x$).

We use the Pythagorean theorem: $a^2 + b^2 = c^2$.

Solution

1. Set up the equation: Let $x$ be half the length of the chord. $x^2 + d^2 = r^2$ $x^2 + 5^2 = 7^2$

2. Solve for $x$: $x^2 + 25 = 49$ $x^2 = 49 - 25$ $x^2 = 24$ $x = \sqrt{24}$

3. Simplify the radical: $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ cm.

4. Find the full length of the chord: Since $x$ is only half the chord, the full length is $2x$: Length $= 2 \times (2\sqrt{6}) = 4\sqrt{6}$ cm.

Conclusion

The length of the chord is $4\sqrt{6}$ cm. Comparing this to the given options, it matches option A.

Exam Tip

Whenever you see a problem involving a chord and the distance from the center, immediately sketch a right-angled triangle. Remember that the radius is always the hypotenuse. If you forget the Pythagorean theorem, just remember: (Half-chord)² + (Distance)² = (Radius)².