Circle area from two-point radius: In the xy-plane, points P(2, 0) and Q(5, 4) are given. If a circle has radius equal to the distance PQ, find its area (use π as pi).

Introduction / Context:
The question combines distance formula in the coordinate plane with the area formula for a circle. First compute the segment length PQ; that length becomes the radius r. Then evaluate area = π * r^2.

Given Data / Assumptions:

• P = (2, 0)
• Q = (5, 4)
• Area of circle = π * r^2 with r = PQ

Concept / Approach:
Use r = sqrt[(x2 − x1)^2 + (y2 − y1)^2]. Then square r to compute area. This avoids working with square roots at the end since r^2 is needed directly in the area formula.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 16 π, 14 π, 32 π: These correspond to incorrect computations of r^2; only 25 π matches Δx^2 + Δy^2 = 25.

Common Pitfalls:

• Using r instead of r^2 in the area formula or mixing up Δx and Δy.

Final Answer: