Chord–distance relation – find radius: In a circular lawn, a straight path of length 16 m forms a chord that is 6 m away from the center. Find the radius of the lawn.

Introduction / Context:
A chord of a circle relates to the radius and the perpendicular distance from the center. The half-chord forms a right triangle with the radius and the distance from the center to the chord.

Given Data / Assumptions:

• Chord length L = 16 m ⇒ half-chord = 8 m.
• Distance from center to chord d = 6 m.
• Radius r is unknown.

Concept / Approach:
Use the right triangle with legs d and half-chord, and hypotenuse r: (half-chord)^2 + d^2 = r^2.

Step-by-Step Solution:
8^2 + 6^2 = r^2 ⇒ 64 + 36 = r^2 ⇒ r^2 = 100.r = 10 m.

Verification / Alternative check:
Reverse: With r = 10 and d = 6, half-chord = √(r^2 − d^2) = √(100 − 36) = √64 = 8 ⇒ full chord 16 m.

Why Other Options Are Wrong:
6 m is the given distance, not radius; 8 m/16 m are chord-related values, not the radius.

Common Pitfalls:
Using diameter in place of radius in the Pythagorean step or forgetting to halve the chord.

Final Answer:
10 m