Constructions: Parallel Lines Etc
Compass and straightedge construction rules
Here are the only things you may do in a construction:
- Draw segments with the straightedge (extending segments is ok)
- Measure distances with the compass
- Draw an arc or circle with a compass
- Determine a point where lines and/or arcs meet
Here are a few of the many things not to do:
- Do not measure anything with a ruler or protractor. These are, even in principle, approximate
- Do not find the point shared by a circle or arc and a tangent to it. (A circle and its tangent are so close near the intersection point, it is hard to find the exact intersection.)
Analogous GeoGebra rules
If you want to try to stay as close as possible to compass and straightedge construction rules, here are the legal moves:
- Draw lines containing two points and segments joining two points
- Draw circles ("circle with center through point" and "circle with radius". For the latter, analogous to measuring a distance with a compass and then drawing a circle of that radius, use distance[A,B] as the distance between A and B.
- Pick a random point or a point where lines and/or arcs meet using the new point tool
- Once you know how to do a given construction, you may want to allow yourself to use the GeoGebra tool that does the same thing, just for convenience.
Relaxed GeoGebra rules
Anything goes!
The constructions
| First, play! | |
| Given a line AB and an intersecting line CB, construct a line containing point C parallel to line AB using what we know about alternate interior angles to force the line to be parallel. | |
| Given two line segments AB and BC, construct a parallelogram that has those segments as two of the sides. | |
| Given two segments, construct a rectangle whose height and length equal the lenghs of those two segments. | |
| Construct two segments that intersect (at an angle not a right angle) and bisect each other. Then construct the parallelogram whose diagonals are those segments. | |
| Given angle ABC, construct a parallelogram containing sides AB and BC by constructing the supplement to angle ABC at point A or point C. | |
| Construct two congruent segments, AB and CD, that bisect each other. Construct the rectangle ABCD whose diagonals are those segments. Then construct a circle that circumscribes ABCD, i.e. points A, B, C and D lie on the circle. | |
| Given two segments, construct a kite that has two legs congruent to the first segment and two legs congruent to the second. | |
| Given a segment AB, construct a perpendicular bisector CD. Then construct kite ACBD. | |
| Given a segment AB, construct a perpendicular bisector CD, not congruent to segment AB for which AC is also a perpendicular bisector. Then construct rhombus ACBD. | |
| Given a circle, inscribe a square in it (so the vertices of the square lie on the circle). | |
| Given obtuse angle ABC, construct an isosceles trapezoid containing vertices A, B, C and one more point. |