Final answer:
To find the area of a circle with diameter endpoints A(1,3) and B(6,15), you calculate the diameter with the distance formula, then find the radius and use the area formula A = πr². The area of the circle is 42.25π units².
Step-by-step explanation:
To determine the area of a circle given the endpoints of its diameter, we first need to calculate the length of the diameter using the distance formula, which is derived from the Pythagorean theorem. For two points A(x1, y1) and B(x2, y2), the distance formula is:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
For points A(1,3) and B(6,15), the diameter is:
Diameter = √[(6 - 1)^2 + (15 - 3)^2] = √[5^2 + 12^2] = √[25 + 144] = √169 = 13
Since the diameter is 13 units, the radius (r) is half of that, which is 6.5 units.
Now, using the formula for the area of a circle (A = πr²), we find:
Area = π(6.5 units)² = π(42.25 units²)
Since the actual calculation to exact decimals can be extensive, we keep the area of the circle in terms of pi:
Area = 42.25π units²