You are right. Perpendicular bisectors are important prior knowledge for constructions. Understanding that a perpendicular bisector of a segment (side of a polygon, side of an angle maybe) bisects the segment at the midpoint, create two congruent, smaller segments is important. Understanding that any point on that bisector is ‘equidistant to both endpoints of the segment – REALLY important. Also, knowing that the angles formed at the intersection of the perpendicular bisector are all right angles (definition of perpendicular) is obviously key too.
As my two previous posts this week have highlighted, circles, particularly the understanding that a circle creates infinite congruent segments (radii) and that circles made from the same radius (distance between center and a point on the circle) are congruent, are foundational understandings needed to do precises geometric constructions (compass and straight edge, NO measurement). Perpendicular bisectors and their properties are another crucial element to many geometric constructions, such as perpendicular lines/segments/rays (that are not bisectors). If students have the understanding of circles and radii properties and perpendicular bisectors, they conceivably can construct, using a straight edge and compass, any 2D geometric construct. (Now, I am not discounting other tools such as patty paper and dynamic geometry software, but I am really focused on why the circle and straight edge are really tools that really help students build relationships between and among geometric constructs).
As an example of the importance of foundational understanding of perpendicular bisector in constructions, I want to discuss the steps needed to construct a perpendicular to a line/segment/ray from a point NOT on that line/segment/ray. This is actually done AFTER learning about constructing perpendicular bisectors of segments (see figure 1 above), because you need that understanding (two congruent circles (made from the compass open to the same radius), using each end point of a given ‘segment’, that intersect, and then connect the intersections) to make a perpendicular.
Basically, if you don’t have a point on the line/segment/ray, you need to ‘create’ a segment on the given line/segment ray) that has endpoints equidistant from that given point. This is where the understanding of circle comes into play – if we use the point as the center of a circle, then have that circle intersect the given line/segment/ray in two points, we automatically know those two points must be equidistant from the given point because they are on the circle, therefore ‘radii’ of the circle if we connected them with a segment (Figure 2).
Once you have the segment that lies on the original line/segment/ray, you use those as endpoints, and follow the process to create a perpendicular bisector, using the two intersection points as the centers of two circles who both have the radius from the end point to the given point NOT on the line/segment/ray. How do we know these are congruent? They are circles with congruent radii or distances from the center (in Fig 3 – compass/circle with radii EC and FC, centers at E and F)(this matches the process from Figure 1).
Finally, using the two congruent circles (green in Figure 4), you will note they intersect at the original given point that was not on the original line/segment/ray. Construct the other intersection. Connect the two intersections with your straight edge, and voila, you have a perpendicular. Which – happens to be the perpendicular bisector of the ‘segment’ we created on the original line/segment/ray (in Figure 4, EF), but is also a perpendicular in general to the entire original line/segment/ray (in our example, AB). In order to create the perpendicular, we NEEDED a perpendicular bisector and it’s properties of equal distance from endpoints of a bisected segment (EF) to create a general perpendicular.
It’s hard to show in images, so I have created a short video walking through the importance of the perpendicular bisector to help you create general perpendiculars to lines/segments/rays. This is a very important construct, especially when thinking about constructing rectangles and squares where perpendicular sides are crucial.
Video: Constructing Perpendicular Bisectors and Perpendiculars (Circles, Radii, and Properties)
The tool being used in these mini-math lessons is the FREE web-based math software, ClassPad.net.
Remember – if you want to save and/or modify any of these activities, create a free account. Some useful links below:
- How to create a free ClassPad.net account (for anyone – students, parents, teachers)
- How to organize activities & share activities
- Some Ready-to-use activities that are already out there
- ClassPad.net YouTube Channel