Final answer:
The coordinates of point B are (-4, -1), and point C is at (2, 1). The locations were determined using the Pythagorean theorem and the properties of slope.
Step-by-step explanation:
The task is to determine the coordinates of point B and C, where points A, B, and C form a right triangle on a coordinate plane. Given that point A is located at (-2,5), the distance between point A and B is the square root of 40, and point B is located 6 units down from point A and to the left. This implies that the y-coordinate of point B is 5 - 6 = -1. The x-coordinate can be found by using the Pythagorean theorem (noting that the x-axis distance is one leg of the right triangle and the y-axis distance is the other).
Since the distance from point A to B is √40 and we know one leg is 6 units (the movement down the y-axis), we can calculate the x-axis leg using the Pythagorean theorem as follows:
√(6² + b²) = √40
36 + b² = 40
b² = 4
b = 2
Since point B is to the left of point A, we subtract 2 from the x-coordinate of point A to get the x-coordinate for point B: -2 - 2 = -4. Therefore point B is at (-4, -1).
For point C, it is located 4 units to the right of point A, which means its x-coordinate is -2 + 4 = 2. The slope between point B and C is -1/3, and since we know the x-coordinate difference between point B and C is 2 - (-4) = 6, we can calculate the change in y (rise) by multiplying the slope (-1/3) by the run (6), which is -2. Hence, the y-coordinate of point C is -1 (from point B) minus the rise of -2, resulting in a y-coordinate of -1 + 2 = 1. Thus, point C is at (2, 1).