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Answer:
(A) see below
(B) the right angle is at vertex B(-4,11)
(1) 3x -4y = -46
(E) the midpoint is (-4, 8.5)
Step-by-step explanation:
(A) The slope of line AB is Δy/Δx = (11-7)/(-4-(-6)) = 4/2 = 2. The slope of line BC is (10-11)/(-2-(-4)) = -1/2. The slopes of AB and BC have a product of 2(-1/2) = -1, so are the slopes of perpendicular lines. The points are distinct, and lines joining two pairs of them are at right angles, so the points form a right triangle.
(B) Point B(-4,11) is the point of intersection of perpendicular segments AB and BC, so is the location of the right angle.
(1) The slope of the hypotenuse AC is ...
Δy/Δx = (10-7)/(-2-(-6)) = 3/4
In point-slope form, the equation for the line through point A with this slope is ...
y -7 = 3/4(x +6) . . . . point-slope form of the equation of the hypotenuse
4y -28 = 3x +18 . . . . multiply by 4
-46 = 3x -4y . . . . . . subtract 18+4y
3x -4y = -46 . . . . . . . standard form equation of the hypotenuse
(E) The midpoint of the hypotenuse is the average of the endpoint coordinates:
M = (A + C)/2 = (-6-2, 7+10)/2 = (-4, 8.5)
The midpoint of the hypotenuse is M(-4, 8.5).
