SOLUTION: This problem is not in my book, my teacher likes to make up his own questions. I can't find anything on perpendicular bisector. I think it may have something to do with midpoint b

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Question 82158: This problem is not in my book, my teacher likes to make up his own questions. I can't find anything on perpendicular bisector. I think it may have something to do with midpoint but I am not for sure. I would attempt it but I don't know where to begin...
Find an equation for the perpendicular bisector of the line segment IJ for the points I(3,5) and J(7,-3)
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
First find the midpoint of the segment with the endpoints (3,5) and (7,-3)

Solved by pluggable solver: To find midpoint of segment connecting two point
The Coordinates of mid point of a line segment joining two points can be calculated using following formulas.

X coordinate of mid point is

X%5Bmid%5D=+%28X+coordinate_of_first_point+%2B+X+coordinate_of_first_point%29%2F2


X%5Bmid%5D+=%283%2B7%29%2F2


X%5Bmid%5D+=%283%2B7%29%2F2=5


Y coordinate of mid point is

Y%5Bmid%5D=+%28Y+coordinate_of_first_point+%2B+Y+coordinate_of_first_point%29%2F2


Y%5Bmid%5D+=%285%2B-3%29%2F2


Y%5Bmid%5D=%285%2B-3%29%2F2=1

Hence, The mid point of segment joining two point (3,5) and (7,-3) is (5,1)



So we know that the bisecting line will go through the point (5,1). Now find the slope of the line going through (3,5) and (7,-3)

Solved by pluggable solver: Finding the slope
To find the slope going from (3,5) to (7,-3) we are going to calculate the change in y over the change in x, or the rise over the run. The change is the difference between the two coordinates. So if the y-coordinate of a point goes from 5 to -3, the change in these numbers is -8 (since -3-5=-8). If the x-coordinate changes from 3 to 7, then the change is 4 (since 7-3=4). So to calculate the slope we use this formula:
Slope:

m=%28change_in_y%29%2F%28change_in_x%29=rise%2Frun where m is the slope

So now we let y%5B2%5D=-3,y%5B1%5D=5,x%5B2%5D=7,x%5B1%5D=3Now plug these numbers into the slope formula:

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29=%28-3-%285%29%29%2F%287-%283%29%29+=+-8%2F4


So after simplification the slope is m=-2



Since the slope of the line through (3,5) and (7,-3) is m=-2 we know the perpendicular slope is
m%5Bp%5D=-1%2Fm where m%5Bp%5D is the perpendicular slope

m%5Bp%5D=-1%2F-2=1%2F2

So the bisecting line has a slope of 1/2 and goes through the point (5,1). So lets find the equation of the line:

Solved by pluggable solver: FIND a line by slope and one point

What we know about the line whose equation we are trying to find out:

  • it goes through point (5, 1)

  • it has a slope of 0.5



First, let's draw a diagram of the coordinate system with point (5, 1) plotted with a little blue dot:

%0D%0A++++drawing%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+++++++++++++grid%281%29%2C%0D%0A+++++++++++++blue%28+circle%28+5%2C+1%2C+0.2+%29+%29+%29%0D%0A++

Write this down: the formula for the equation, given point x%5B1%5D%2C+y%5B1%5D and intercept a, is

y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29 (see a paragraph below explaining why this formula is correct)

Given that a=0.5, and system%28+x%5B1%5D+=+5%2C+y%5B1%5D+=+1+%29+, we have the equation of the line:

y=0.5%2Ax+%2B+-1.5

Explanation: Why did we use formula y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29 ? Explanation goes here. We are trying to find equation y=ax+b. The value of slope (a) is already given to us. We need to find b. If a point (x%5B1%5D, y%5B1%5D) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (x%5B1%5D, y%5B1%5D): y%5B1%5D+=+a%2Ax%5B1%5D%2Bb Here, we know a, x%5B1%5D, and y%5B1%5D, and do not know b. It is easy to find out: b=y%5B1%5D-a%2Ax%5B1%5D. So, then, the equation of the line is: +y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+.

Here's the graph:

%0D%0A++++drawing%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+++++++++++++grid%281%29%2C%0D%0A+++++++++++++graph%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+0.5%2Ax%2B-1.5+%29%2C%0D%0A+++++++++++++blue%28+circle%28+5%2C+1%2C+0.2+%29+%29%2C%0D%0A+++++++++++++locate%28+5%2C+1%2C+%27%285%2C+1%29%27+%29+%29%0D%0A+++++++++++++%29%0D%0A++++++++++


So the equation of the bisecting line is

y=%281%2F2%29x-3%2F2 or y=0.5x-1.5