Question

The coordinates of the vertices of AHIJ are H(-3,3), I (1,-3) and J(3,7). Using coordinate geometry prove

that AHIJ is an isosceles right triangle.

Answer 1

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.