Final answer:
The area of the circle with diameter endpoints A(2, 1) and B(5, 5) is 25π square units. The diameter is calculated using the distance formula, and the radius is half the diameter.
Step-by-step explanation:
To find the area of the circle in terms of pi, given the endpoints of a diameter A(2, 1) and B(5, 5), we first need to calculate the length of the diameter using the distance formula. Then we can find the radius and use the formula for the area of a circle, A = πr².
The distance formula is given by d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points. Here, (x1, y1) = (2, 1) and (x2, y2) = (5, 5), so the diameter is √((5 - 2)² + (5 - 1)²) = √(9 + 16) = √25 = 5 units. The radius r is half the diameter, so r = 5 / 2 units. Substituting the radius into the area formula, we get A = π(5 / 2)² = π · 2.5² = π · 6.25 = 25π square units. Therefore, the correct answer is C. 25 π square units.