Final answer:
The correct points for A and B in Triangle ABC, with C being (6,2) and AC having a slope of 3, are A at (-12,8) and B having the same y-coordinate as A, which is 8. Point E (7,5) is incorrect as it would not form a horizontal line with C.
Step-by-step explanation:
In the context of Triangle ABC being a right triangle with point C at (6,2) and segment AB being horizontal, the slope of segment AC equal to 3 means for every horizontal change of 1, there’s a vertical change of 3.
Looking at the given options, we are seeking a point A that, when connected to point C, will form the correct slope of 3. Let's calculate the slope using two points (x1, y1) and (x2, y2) with the formula (y2 - y1)/(x2 - x1).
For a horizontal segment AB, the y-coordinate must be the same for points A and B. Therefore, we can immediately eliminate any points that do not share the y-coordinate of point C, which is 2.
Thus, point E (7,5) cannot be correct as it does not form a horizontal line with point C.
Examining the remaining options and applying the slope formula with point C, we calculate the slope for each potential point to find which one yields a slope of 3. The point that, when paired with (6,2), gives a slope of 3, will be the correct choice for point A.
Additionally, as segment AB is horizontal, we can conclude that point B must have the same y-coordinate as point A.
After verifying the slopes, we find that the only pair that produces the proper slope is point A as (-12,8) and, accordingly, since AB is horizontal, point B must have a y-coordinate of 8 and can be located to the left of A to make triangle ABC.