The largest chord of a circle has length 20.4 cm. What is the radius of the circle?

Introduction / Context:
This question relies on an important and very simple property of circles: the largest possible chord in any circle is its diameter. Once you recognize that, finding the radius from the length of the largest chord becomes an easy one step calculation.

Given Data / Assumptions:

• The largest chord of the circle has length 20.4 cm.
• The largest chord in a circle is always the diameter.
• We must find the radius of the circle.
• Radius is half of the diameter.

Concept / Approach:
Any chord of a circle is a straight line segment joining two points on the circle. The chord that passes through the centre is called the diameter and is the longest chord. No other chord can be longer than the diameter. Therefore, if the question explicitly tells us that the largest chord has a certain length, that length must equal the diameter. Once we identify this, we simply divide by 2 to obtain the radius.

Step-by-Step Solution:
Step 1: Recognize that the largest chord in a circle is the diameter. Step 2: Therefore, the given length 20.4 cm is the diameter of the circle. Step 3: The radius r is half the diameter: r = diameter / 2. Step 4: Compute r = 20.4 / 2 = 10.2 cm. Step 5: Hence, the radius of the circle is 10.2 cm.

Verification / Alternative check:
Consider any circle with radius r. The diameter is 2r, and any other chord lies inside the circle, forming a right triangle when you draw a radius to its midpoint. This right triangle geometry ensures that all other chords are shorter than 2r. Therefore, if 20.4 cm is indeed the largest chord, it must be the diameter, and the radius must be exactly 10.2 cm, with no ambiguity.

Why Other Options Are Wrong:
If the radius were greater than 10.2 cm, then the diameter would be greater than 20.4 cm, which contradicts the statement that 20.4 cm is the largest chord. If the radius were less than 10.2 cm, the diameter would be shorter than 20.4 cm, meaning a chord of length 20.4 cm could not even exist in that circle. Saying the radius is greater than or equal to 10.2 cm or that it cannot be determined conflicts with the unique relationship between the largest chord and the diameter in a circle.

Common Pitfalls:
One mistake is to overthink the problem and look for complicated geometry when a simple property suffices. Another is to confuse the largest chord with a random chord and think there might be many possibilities for the radius. Remembering that only the diameter is the maximum chord length allows you to immediately convert chord length to radius by halving it.

Final Answer:
The radius of the circle is 10.2 cm.