A single straight line cuts two concentric circles (same center). The chord lengths formed on the inner and outer circles are 4 cm and 16 cm, respectively. Find the difference between the squares of their radii (in cm²).

Introduction / Context:When a line at distance d from the center cuts a circle of radius r, the chord length L satisfies L = 2 * √(r^2 − d^2). With two concentric circles cut by the same line, d is common to both, enabling an easy elimination approach to find r2^2 − r1^2.

Given Data / Assumptions:

• Inner circle chord length L1 = 4 cm.
• Outer circle chord length L2 = 16 cm.
• Both chords arise from the same line, so the perpendicular distance d from center to the line is the same.

Concept / Approach:Use the relation (L/2)^2 = r^2 − d^2 for each circle, then subtract to eliminate d^2 and directly obtain the difference of squared radii.

Step-by-Step Solution:

Verification / Alternative check:Pick any d with 0 < d < r1; for example d = 1 gives r1^2 = 5, r2^2 = 65, difference = 60, confirming independence of d.

Why Other Options Are Wrong:

• 240, 120, 90: Each assumes incorrect squaring or arithmetic; only 60 satisfies the chord–radius relation.

Common Pitfalls:Forgetting to halve the chord length before squaring; mixing up r^2 − (L/2)^2 with (r − L/2)^2.

Final Answer:60