The vertices of Triangle MAT are A (–3, 1), M (3, 4), and T (–2, –1). Find the coordinate, H, that would turn triangle MAT into rectangle MATH. Prove that MATH is a rectangle

Question

The vertices of Triangle MAT are A (–3, 1), M (3, 4), and T (–2, –1). Find the coordinate, H, that would turn triangle MAT into rectangle MATH. Prove that MATH is a rectangle

Answer 1

Drawing the figure on the coordinate plane, H would be (4,2)

If all 4 angles are right angles then the figure is a rectangle.

Compute slope of all 4 lines:

AT = (-1-1)/(-2--3) = -2

AM = (4-1)/(3--3) = 3/6 = 1/2

MH = (4-2)/(3-4) = -2

HT = (2--1)/(4--2) = 3/6 = 1/2

When lines are perpendicular their slopes are the negative reciprocals.

So AT perpendicular to AM, AM perpendicular to MH, MH perpendicular to HT and HT perpendicular to AT and thus all 4 angles are right angles.