Final answer:
By calculating the slopes of the sides of the triangle using their coordinates, it is determined that the slopes of gh and jg are negative reciprocals. Therefore, Δghj is a right triangle because it has a pair of perpendicular sides.
Step-by-step explanation:
To determine whether Δghj is a right triangle, we can use the slopes of the sides of the triangle. The slope of a line can be found using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.
The slope of gh¯ can be calculated using the points g(-1, 3) and h(1, 2), yielding a slope of m = (2 - 3) / (1 - (-1)) = -1 / 2.
The slope of hj is calculated using the points h(1, 2) and j(-3, -1), yielding a slope of m = (-1 - 2) / (-3 - 1) = -3 / -4 = 3 / 4.
Finally, the slope of jg is found using the points j(-3, -1) and g(-1, 3), yielding a slope of m = (3 - (-1)) / (-1 - (-3)) = 4 / 2 = 2.
Now, if two sides are perpendicular, their slopes are negative reciprocals of each other (m1 * m2 = -1). Looking at the slopes calculated:
- slope of gh¯ = -1 / 2, and slope of jg = 2, we have (-1 / 2) * 2 = -1, indicating gh¯ is perpendicular to jg.
Thus, Δghj is a right triangle because it has a pair of perpendicular sides (gh¯ and jg).