Final answer:
To determine if Δghj is a right triangle, calculate the slopes of sides gh, hj, and jg. The slope of gh is -1/2, hj is 3/4, and jg is 2. Δghj is a right triangle because gh and jg have slopes that are negative reciprocals, indicating they are perpendicular.
Step-by-step explanation:
To determine if Δghj is a right triangle, we will calculate the slopes of the sides gh, hj, and jg. The slope of a line between two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1) / (x2 - x1).
Slope of gh
The coordinates of the vertices g(-1, 3) and h(1, 2). Applying the slope formula, we get m = (2 - 3) / (1 - (-1)) = (-1) / (2) = -1/2. So the slope of gh is -1/2.
Slope of hj
The coordinates of the vertices h(1, 2) and j(-3, -1). Applying the slope formula, we get m = (-1 - 2) / (-3 - 1) = (-3) / (-4) = 3/4. So the slope of hj is 3/4.
Slope of jg
The coordinates of the vertices j(-3, -1) and g(-1, 3). Applying the slope formula, we get m = (3 - (-1)) / (-1 - (-3)) = 4 / 2 = 2. So the slope of jg is 2.
To determine if Δghj is a right triangle, we check if any two slopes are negative reciprocals of each other. In this case, the slopes of gh (-1/2) and jg (2) are negative reciprocals. Therefore, Δghj is a right triangle because the slopes indicate that gh and jg are perpendicular to each other.