Algebraic Proof of the Perpendicular Bisector Theorem using Coordinate Geometry
Lesson 2 of 7
Objective: SWBAT use algebra to verify that any point on the perpendicular bisector of a given segment is equidistant from the endpoints of the segment.
Activating Prior Knowledge
This lesson is algebra-based and I don't want to assume that my students have all of the necessary algebra skills at the ready. This section gives them an opportunity to warm up the skills.
First we sketch the perpendicular bisector of a segment to make sure that students know what the perpendicular bisector is and what it does.
Next we practice writing linear equations (a) through a point with a given slope, and (b) through a point perpendicular to a given line.
Finally we practice squaring binomials (without having to distribute) and finding distance in the coordinate plane.
Some of my students will be able to work through this activity quickly on their own. Others will need to have things demonstrated for them. Typically, what I'll do is give the class enough time to work on #1, for example, then I'll demonstrate #1 for anyone who needs to see it. Then I continue in this fashion until I've demonstrated all of the problems.
Students work with their A-B partners to complete Perpendicular Bisector Theorem. As I'm walking around checking on students, I provide just-in-time feedback. After teaching this lesson several times, I know the following are likely Points of Intervention:
- Deciding which point(s) to use to write the equation of the perpendicular bisector. (Questions: I see that you used the coordinates of one of the endpoints...does the perpendicular bisector pass through the endpoints of the segment?; What point does the perpendicular bisector intersect by definition?; How do you find the coordinates of the midpoint?
- Deciding which slope to use. (Questions:You've calculated the slope of segment AB...is that the same as the slope of the perpendicular bisector?; What does the "perpendicular" in perpendicular bisector mean?... Perpendicular to what?; How are the slopes of perpendicular lines related to each other?)
- Showing that a point is equidistant from two endpoints. (Questions: What is it that we're trying to show?; What does equidistant mean?; Which are the endpoints of segment AB? What tool do we have that could show equidistance given only coordinates?
- Finding the coordinates of P (Questions: What does generic mean?; On the perpendicular bisector, If x = 2, then what does y equal? On the perpendicular bisector, If x = 5, then what does y equal?...What computations did you perform on each x value I gave you in order to get the corresponding y value? So, in general, how do you express doing those same computations to a generic x? ) (Feedback: When students want to fill in the blank with y, I explain that this is not wrong, nor is it useful since (x,y) could be any point in the coordinate plane and does not specify a point on the perpendicular bisector. x can be any number, but the corresponding value of y is related to the chosen value of x)
- Problems putting (x, 3x+4) into the distance formula (Feedback: Remember that we are showing that any point (x,y) that satisfies y=3x+4 is equidistant from (-2,8) and (4,6), so don't expect to get a numeric answer. You'll get an algebraic answer, and you'll have to determine whether or not it is proof that any point on the perpendicular bisector is equidistant from the endpoints of the segment it bisects.
Once students have had enough time to complete the handout, I call their attention to the front of the class so that I can recap the major concepts and procedures from the lesson. First, I remind students that when we want to write the equation of a line, we usually need a point and a slope. In this case, with the line being the perpendicular bisector, our point is the midpoint and our slope is the opposite reciprocal of the segment's slope.
Next I discuss how we chose specific points on the perpendicular bisector to see if they were equidistant from A an B, the endpoints of the segments. I explain that even if we had chosen a million specific points on the perpendicular bisector and they all were equidistant from the endpoints of the segment, that would not constitute proof of the perpendicular bisector theorem. Next I ask students to discuss with their partners the purpose of introducing the generic point P. Then I explain that P represents ALL points on the perpendicular bisector (infinitely many of them) and how that which is true for P must be true for all points on the perpendicular bisector. Therefore if we prove that P is equidistant from the endpoints of the segment then all points on the perpendicular bisector must be equidistant from the endpoints of the segment.
Finally, I show the algebra required to show that P is equidistant from A and B. That work can be seen on the Perpendicular Bisector Theorem_model response.
Before students leave, I ask them to respond to the following prompt:
What were we attempting to show in this lesson? Explain in detail how we went about showing it. What were our findings?
When I read these responses, it will let me know how well students have seen the coherence in the lesson and how well they have understood the reasons for the individual steps we took.