Final answer:
The triangle with slopes 3/4, -1/2, and 2 cannot be a right triangle because there is only one pair of perpendicular sides indicated by the slopes -1/2 and 2, while a right triangle requires exactly two pairs of sides with slopes that are negative reciprocals of each other.
Step-by-step explanation:
To determine if the triangle with slopes of its sides being 3/4, -1/2, and 2 is a right triangle, we can use the concept of perpendicular slopes. In a right-angled triangle, the slopes of the perpendicular sides are negative reciprocals of each other. That is, if one slope is m, the slope of the perpendicular side is -1/m.
Considering the given slopes, let's find the negative reciprocals:
- The negative reciprocal of 3/4 is -4/3.
- The negative reciprocal of -1/2 is 2/1 or simply 2.
- The negative reciprocal of 2 is -1/2.
As we can see, the slope of -1/2 has a negative reciprocal of 2, which matches one of the given slopes, which implies these two sides are perpendicular. However, to form a right triangle, we need another pair of slopes that are negative reciprocals of each other. Since there is no slope that is a negative reciprocal of 3/4 among our given slopes, the given triangle cannot be a right triangle because there is only one pair of perpendicular sides, whereas a right triangle requires exactly two.