The Perpendicular Bisector
*The perpendicular bisector of a line segment is the line that is perpendicular to the line segment, and passes through its midpoint. In other words, the perpendicular bisector meets the line segment at a 90 degree angle, and divides it evenly in half. REMEMBER: The perpendicular bisector does not necessarily travel through the vertex. When calculating the equation of the perpendicular bisector, always substitute in the coordinates of the midpoint of the line which the perpendicular bisector is dividing.
* The circumcentre is the point of intersection of all three perpendicular bisectors of a circle. It can be determined by finding the equations of the perpendicular bisectors of two sides, then finding the point of intersection of those two lines.
SLOPES
*The slope of the perpendicular bisector is the negative reciprocal of the line segment which it is dividing.
To find the slope of the perpendicular bisector, follow these steps.
1. Calculate the slope of the line segment that the perpendicular bisector will be dividing by using the slope formula:. The points you will substitute into (x1,y1) & (x2,y2) are the coordinates of the end points of the line segment.
2. Take the negative reciprocal of the slope. For example, if the slope of the first line segment is 5/4, the slope of the perpendicular bisector will be -4/5.
You now have the slope of the perpendicular bisector.
EQUATION
To find the equation of the perpendicular bisector, follow these steps:
1. Find the midpoint of the first line segment (for instructions on how to do this, see the Midpoint Page.
2. Find the slope of the perpendicular bisector. (Instructions are above).
3. Find the y-intercept of the perpendicular bisector. To do this, you need the slope of the perpendicular bisector (found in step 2) and a point on the line. To find a point on the perpendicular bisector line, simply substitue in the coordinates of the midpoint of the line which the perpendicular bisector is dividing into the y+mx+b formula.
4. Now that you have calculated the slope and the y-intercept of the perpendicular, substitute these values one last time into the y=mx+b formula to determine the equation of the perpendicular bisector.
You now have the equation of the perpendicular bisector.
Practice Questions
Page 174 #15.
Page 195 #10, 14, 17c, 26.
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