Constructing perpendiculars

Topics in this lesson

Constructing the perpendicular bisector of a segment
- Constructing the midpoint of a segment
- Constructing a perpendicular to a line through a given point
Copying a length
Constructing a regular tessellation of the plane by squares

The perpendicular bisector of a line segment

Definition: The perpendicular bisector of a segment is the line that is perpendicular (at a right angle) to the segment and goes through the midpoint of the segment.

Theorem: The perpendicular bisector of a segment is the set of all points that are the same distance from both the endpoints of the segment.

Note: A theorem is a mathematical fact that has been proved from more basic facts. More about this later.

Constructing the perpendicular bisector of a segment

Draw a circle whose center is one of the endpoints of the segment, and whose raduis is more than half the length of the segment.
Draw another circle with the same radius, whose center is the other endpoint of the segment.
Draw the line through the two points where the circles intersect.

Note: You don't have to draw the entire circle, but just the arcs where the two circles intersect.

Spinoff: Construct the midpoint of a segment.

Construct the perpendicular bisector. The midpoint of the segment is where the perpendicular intersects the line (by definition).

Axiom: Given a line and a point, there is one and only one perpendicular to the line through the point.

Note: An axiom is the most basic type of mathematical fact. Axioms are the starting rules which are just assumed, not proved. (You have to start somewhere.) The first theorems are proved from definitions and axioms.

Spinoff: Construct the perpendicular to a line through a given point.

The main idea is to construct an appropriate line segment on the line, then construct the perpendicular bisector of this segment.

Construction steps

A. If the point is not on the line,
draw a circle whose center is the given point, and whose radius is large enough so that the circle and line intersect in two points, P and Q.
B. If the point is on the line,
draw a circle whose center is the given point; the circle and line intersect in two points, P and Q.
Construct the perpendicular bisector of segment PQ.

Copy a length

You are given a line segment, and would like to construct another segment of the same length at a given point on another line.

Construction steps

You are given a line segment AB with length L, and would like to construct another segment of the same length at a given point C on another line.
Use the endpoints A and B to set your compass opening. Draw a circle with that raduis and center C. Pick one of the intersection points of the line and the circle as one endpoint; C is the other endpoint.

Note: You don't have to draw the entire circle, but just the arc where the line and circle intersect.

Construct a tessellation of the plane by squares

Make some square graph paper (a tessellation of the plane by squares) using only straightdge and compass. Follow these steps:

Draw a line.
Set your compass opening to the length you want for the side of a square.(Use at least 1 inch, or it will be too small to draw accurately.)
Draw a circle with that radius and center on the line.
Draw more circles with the same radius whose centers are the points where the last circle intersected the line.
Continue drawing circles along the line.
For each circle center, draw a perpendicular to the original line through that point.
Mark off the same length (radius) on each of the perpendicular lines, starting from the circle centers.
Draw lines parallel to the original lines using the marks you just made.

Group problems

Consider the equilateral triangle tessellation constructed in the first class. You may use the points of intersection of all the circles, and any lines through those points. Is it possible to construct a right angle with these points and lines? Why or why not? Is it possible to construct a square with these points and lines? Why or why not?

Homework problem

Invent a method to construct a single square with straightedge and compass, using as few steps as possible. Do the construction and write clear instructions.

Definition: A median of a triangle is a line segment that goes from a vertex to the midpoint of the opposite side.
Construct two random triangles (one with an obtuse angle). Make them big: one per page isn't too big. For each triangle,
1. Construct the perpendicular bisectors of all three sides.
2. Construct all three medians of the triangles (on the same drawing; use a different color.)
3. Make a conjecture about the perpendicular bisectors of the sides of any triangle.
4. Make a conjecture about the medians of any triangle. (Hint: How could three line segments intersect?)

Extensions (group or individual)

Prove that the first construction in this lesson really does construct the perpendicular bisector. Use any definitions and theorems given so far. Remember that two points determine a line.
How do you know that the perpendicular bisector goes through the original point C, and is perpendicular to the original line?
How do you know that that the new segment in the copy length construction has the same length as the old segment?
Choose one square in the graph paper you constructed and prove that all four of its sides have equal length.