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.
| 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, and center 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.
GP1. How do you know that this line cuts the original segment in half, and is perpendicular to it?
Axiom: Given a line and a point, there is one and only one perpendicular to the line through the point.
The main idea is to construct a line segment on the line, then construct the perpendicular bisector of this segment.
| 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. |
GP2. How do you know that the perpendicular bisector goes through the original point C, and is perpendicular to the original line?
You are given a line segment, and would like to construct another segment of the same length at a given point on another line.
| 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.
GP3. How do you know that that the new segment has the same length as the old segment?
IP0. Invent a method for constructing a square and write instructions for your method.
IP1. Make some square graph paper (a tessellation of the plane by squares) using only straightdge and compass.
Definition: An isosceles triangle is a triangle two of whose sides are equal.
Definition: Two figures are congruent if they have the same shape and size.
Theorem: If the lengths of corresponding sides of two triangles are equal, then the triangles are congruent.
The main idea is to make an isosceles triangle out of the angle, then copy the triangle by copying the lengths of its sides.
| You are given an angle ABC, and would like to construct another angle with the same measure, with one side on a given line, and vertex at a given point D. |  |
| Draw a circle whose center is B; Call the intersection points of the circle and the rays of the angle P and Q. Draw segment PQ to get triangle BPQ. |  |
| Use the same radius to make a circle with center at D on the line. This copies side BQ to side DQ' of the new triangle. Becasuse the length BP is the same as BQ, the copy of the point P will also be on the new circle. |  |
| Draw a circle with radius the length of PQ and center Q'. The copy of the point P will also be on this second circle. |  |
| Pick one of the intersection points of the two circles as P'. Then angle P'DQ' is a copy of angle PBQ. |  |
GP4. How many different positions could triangle P'DQ' have? (Its position depends on several choices that were made during the construction.)
The main idea is to make an isosceles triangle out of the angle as above, then construct the perpendicular bisector of the new side.
IP2. What angles can you construct once you know how to construct equilateral triangles, perpendicular lines, and angle bisectors? (There are infinitely many.)
IP3. A regular octagon has 8 angles each measuring 135 degrees. Invent a method for constructing a regular octagon with straightedge and compass. Construct a regular octagon and give instructions.
IP4. Choose two of the patterns below. Reproduce one of them using only straightedge and compass; use square graph paper for the other.
a. 
b. 
c. 
The main idea is to make an angle, and copy it so that one of its sides is parallel to the original line.
| You are given a line m and a point D not on the line, and would like to construct another line parallel to m and containing the point D. |  |
| Choose any points E and F on line m; draw line DE to make an angle DEF. |  |
| Copy angle DEF to vertex D, with one side along line DE. (You must choose the correct position for the copied angle; look at the question after the copying an angle construction.) |  |
GP5. Can you prove that line DF' is parallel to line m? You will have to use some facts not presented in the course so far.