Question

Let A=(-6,7), B=(-4,11), and C=(-2.10) be three points in the coordinate plane (A) Verify that the three points form a right triangle, and Identify the point at the vertex where the right angle is. (Make sure to provide thorough explanation/work to justify your answer.) Lletres (1) Determine the equation of the line on which the hypotenuse of the right triangle lles. (e) Determine the coordinates of the midpoint of the hypotenuse.

Answer 1

(A) see below

(B) the right angle is at vertex B(-4,11)

(1) 3x -4y = -46

(E) the midpoint is (-4, 8.5)

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).