Find the length of a chord 3cm from the centre of a circle of radius 5cm

1. Analyze the question

We are given a circle with a radius of $5\text{ cm}$. A chord is drawn inside this circle, and its perpendicular distance from the center of the circle is $3\text{ cm}$. We need to find the total length of this chord.

2. Explain the concept

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

• The hypotenuse is the radius of the circle ($r = 5\text{ cm}$).
• One leg is the perpendicular distance from the center to the chord ($d = 3\text{ cm}$).
• The other leg is half the length of the chord ($x$).

We can solve for $x$ using the Pythagorean theorem: $a^2 + b^2 = c^2$.

3. Show the solution

• Step 1: Set up the Pythagorean equation. Let $x$ be half the length of the chord. $x^2 + 3^2 = 5^2$
• Step 2: Solve for $x$. $x^2 + 9 = 25$ $x^2 = 25 - 9$ $x^2 = 16$ $x = \sqrt{16}$ $x = 4\text{ cm}$
• Step 3: Find the total length of the chord. Since $x$ is only half the chord, the total length is $2 \times x$. Total length $= 2 \times 4\text{ cm} = 8\text{ cm}$.

4. Identify the answer

The calculated length is $8\text{ cm}$, which corresponds to option A.

5. Tip

Whenever you see a circle problem involving a chord and a distance from the center, immediately visualize a 3-4-5 right-angled triangle. This is a common Pythagorean triple that examiners love to use to save you time during the JAMB exam. If you recognize the numbers 3 and 5, you can quickly deduce the missing side is 4 without doing long calculations!