Final answer:
The perimeter of triangle ABC with the given points is found by adding the lengths of the sides and the correct answer is 9 + √41 units.
Step-by-step explanation:
The perimeter of triangle ABC with points A(-1,5), B(4,5), and C(-1,1) can be found by calculating the lengths of the sides of the triangle using the distance formula and summing them up. The distance formula is √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of two points.
- Length AB: Since A and B have the same y-coordinate, the distance is simply the difference in the x-coordinates, which is |4 - (-1)| = 5 units.
- Length AC: AC is vertical, so the distance is the difference in the y-coordinates, which is |5 - 1| = 4 units.
- Length BC: We apply the distance formula to get √((4 - (-1))² + (5 - 1)²) = √(25 + 16) = √41 units.
Add the lengths of the three sides to get the perimeter: 5 + 4 + √41 = 9 + √41 units. Therefore, the correct answer is (c) 9 + √41 units.