Final answer:
triangle AHIJ is indeed an isosceles right triangle.
Step-by-step explanation:
To prove that triangle AHIJ is an isosceles right triangle, we need to show that two sides are congruent and one angle is a right angle.
1. First, we can find the lengths of the sides using the distance formula.
Distance between points H and I = √((-3 - 1)^2 + (3 - (-3))^2)
= √(16 + 36)
= √52
Distance between points I and J = √((1 - 3)^2 + (-3 - 7)^2)
= √(4 + 100)
= √104
Distance between points J and H = √((-3 - 3)^2 + (3 - 7)^2)
= √(36 + 16)
= √52
2. We can see that the distances between points H and I and between points H and J are equal, so two sides are congruent.
3. To show that the triangle has a right angle, we can calculate the slopes of the sides.
Slope of HI = (3 - (-3))/(-3 - 1)
= 6/(-4)
= -3/2
Slope of HJ = (7 - 3)/(3 - (-3))
= 4/6
= 2/3
Since the slopes are negative reciprocals (-3/2 * 2/3 = -1), the sides HI and HJ are perpendicular and form a right angle at H.
Therefore, triangle AHIJ is an isosceles right triangle.