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a) line: y = x - 11
b) k = -9
c) AP = BP = √74
Step-by-step explanation:
There are four (4) formulas that are needed for solving this problem.
m = (y2 -y1)/(x2 -x1) . . . . . . slope of a line between points (x1, y1) and (x2, y2)
M = ((x1 +x2)/2, (y1 +y2)/2) . . . . midpoint of the segment between the points
y -y1 = m(x -x1) . . . . . point-slope equation of a line given point and slope
d = √((x2 -x1)² +(y2 -y1)²) . . . . distance between two points
In addition, you need to know that perpendicular lines have opposite reciprocal slopes.
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The perpendicular bisector of a segment between two points will be a line with a slope that is the opposite reciprocal of the slope of the segment. That line will go through the midpoint of the segment.
a) The slope of the segment between A(7, -2) and B(9, -4) is ...
m = (-4-(-2))/(9-7) = -2/2 = -1
The midpoint of AB is ...
M = ((7+9)/2, (-2-4)/2) = (8, -3)
The opposite reciprocal of the slope of AB is ...
m = -1/(-1) = 1
Then the point-slope equation of the perpendicular bisector is the line with slope 1 through the point (8, -3):
y -(-3) = 1(x -8)
y = x -11
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b) Point P on the line has x-value of 2, so the y-value will be ...
k = 2 -11 = -9
Point P has coordinates (2, -9).
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c) The length of segment AP is given by the distance formula:
d = √((2 -7)² +(-9-(-2))²) = √((-5)² +(-7)²) = √74
The length of segment BP is likewise given by the distance formula:
d = √((2 -9)² +(-9 -(-4))²) = √((-7)² +(-5)²) = √74
AP = √74 = BP ⇒ AP ≅ BP
