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Question 1166754: Find the equation of the perpendicular bisector of the line segment joining the points (9,7) and (15,1). Write the equation in the form y=mx+b and identify m and b.
Found 2 solutions by solver91311, Theo:
Answer by solver91311(24713)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the equation of a straight line is y = mx + b.
m is the slope.
b is the y-intercept.
the slope is equal to (y2-y1)/(x2-x1)
(x2,y2) and (x1,y1) are points on the line.
let (x1,y1) = (9,7) and let (x2,y2) = (15,1).
slope = (y2-y1)/(x2-x1) = (1-9)/(15-7) = -8/8 = -1
the equation of the given line becomes y = -1*x + b
replace x and y with the value of one of the points and solve for b.
i used (9,7) to get 7 = -9 + b
solve for b to get b = 16
the equation of the given line becomes y = -x + 16
the slope of the line perpendicular to it is equal to the negative reciprocal of the slope of the given line.
the reciprocal of the slope is equal to 1/-1 = -1.
the negative of that is equal to 1.
the equation of the line perpendicular to the given line would be equal to y = x + b.
this perpendicular line would have to go through the midpoint of the given line.
the two points of the given line are (9,7) and (15,1).
these points have been assigned to (x1,y1) = (9,7) and (x2,y2) = (15,1).
the formula for the midpoint of the given line is ((x1+x2)/2,(y1+y2)/2).
replacing with respective values, you get the midpoint of the line is ((9+15)/2,(7+1)/2).
that makes the midpoint equal to (12,4).
the perpendicular bisector of the line has a slope of 1 and goes through the point (12,4).
the general equation of a straight line is y = mx + b
m is the slope and b is the y-intercept.
when m = 1 and (x,y) = (12,4), the equation becomes 4 = 12 + b
solve for b to get:
b = 4 - 12 = -8
the equation of the perpendicular bisector line is therefore y = x - 8.
you have the equation of two lines.
y = -x + 16 which is the equation of the given line.
y = x - 8 which is the equation of the line that is the perpendicular bisector of the given line.
these lines are perpendicular to each other because their slopes are negative reciprocals of each other.
to find the intersection point, solve these 2 equations simultaneously as shown below:
the two equations are:
y = x - 8
y = -x + 16
add the 2 equations together to get:
2y = 8
solve for y to get:
y = 4
replace y with 4 in the first equation to get:
4 = x - 8
solve for x to get x = 12.
since y = 4 and x = 12, the intersection point is at (12,4).
this is what we derived earlier as the midpoint of the given line.
it is therefore also the intersection of both the given line and the line that is the perpendicular bisector of the given line.
the equation of the given line and the line perpendicular to it can be graphed as shown below:
the graph shows the given points of the given line and the point that is the bisector of the given line plus the line that is the perpendicular bisector of the given line.
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