Objective and Overview
In this unit, students make formal geometric constructions with Polypad's geometry tools (utensils) while getting a greater insight into geometrical concepts.
In ancient times, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compasses. Today this tradition can be used to create a series of puzzle-like lessons. This unit focuses on the basics of geometric constructions:
- Copying a segment
- Bisecting a segment
- Finding the perpendicular bisector of a line segment
- Constructing the center of a circle
- Creating perpendicular lines
- Constructing a line parallel to a given line through a point
- Copying an angle
- Bisecting an angle
Share this Polypad with students. Ask them to copy the line segment i.e; to draw a line segment congruent to . Students might notice that the line is locked on Polypad so that are not able to select it and just copy it. Invite students to share their methods. Perhaps many used the ruler.
Then, ask them to do the same thing without measuring it with the ruler. Invite them to share their thinking.
Clarify with the students that it is possible to copy a segment without measuring it. You may use the demo video below or use the construction steps to copy the segment.
Copying means constructing another segment, say , which is congruent to .
- Place the compass point on A and the pencil end on point B.
- Place the compass point on C and draw an arc without changing the compass width.
- Connect any point (D) on the arc with point C. has the same length as as long as the compass width is constant.
Discuss with the students the meaning of the words to sketch, draw, and construct using the copying the segment activity.
In the first activity, they draw the line using a ruler, whereas in the second one they constructed the segment with the compass. Geometric construction means drawing lines, angles, and shapes accurately without using numbers or equations. To do that, we only need two tools! A straight-edge and a compass. Remind students that a straight-edge is like a ruler but without any markings. A compass allows you to draw a circle of a given size around a point.
The Greeks formulated a great portion of geometry over 2000 years ago. In particular, the mathematician Euclid documented geometric concepts, axioms, logic, and constructions in his book “Elements”. Therefore, these constructions are also known as Euclidean constructions. You may learn more about Euclid’s axioms on Mathigon’s Euclidean Geometry Course.
While studying the geometric constructions, students will see the close connection to axiomatic logic used by Euclid to prove his theorems. He proved everything with a minimum of assumptions such as axioms and postulates, and constructed the most complex geometric shapes with a minimum of tools such as a compass and straightedge.
From those days to now, using only a straight-edge and compass to construct geometric shapes became almost like a puzzle for mathematicians. Invite students to join you in following in the footsteps of the great mathematicians! Share the following challenges and puzzles students. Options include presenting them one at a time, sharing them all with students at once, or sharing some with groups.
Bisecting a segment and constructing the perpendicular bisector of a segment
First, remind students that there are two rules:
- They may only use the compass and the straight-edge tools.
- They cannot use the ruler to measure lengths or the protractor to measure the angles.
Share this Polypad with students. Here is the solution:
- Place the compass point on point A and open the compass larger than half of the segment's length to draw an arc.
- Keep the compass width constant, place the compass point on B to draw the arc. Make sure that the arcs intersect at two points. Label the intersection points as C and D.
- Connect C and D with the straight edge. The line segment CD intersects with the segment AB at a point. Label the point as E.
- E is the midpoint (median) of and is the perpendicular bisector of the segment AB.
Center of a Circle
Being able to find the midpoint of a segment without measuring and constructing its perpendicular bisector helps with more complex constructions. One such example is finding the center of a circle.
Share this Polypad with students and ask them to construct its center with a compass and straight edge. Let them discuss the possible properties of a circle that can help them to construct (locate) its center. Here is the solution:
One of the easiest methods is to find the intersection point of the perpendicular bisectors of any two chords of the circle. This method is based on the fact that the perpendicular bisector of a chord passes through the center of a circle.
Share this Polypad to construct parallel and perpendicular lines with students. Here is the solution:
There can be two different options for constructing a perpendicular line to a given line:
- Constructing the perpendicular line from a point that is on the line
- Constructing the perpendicular line from a point that is not on the line
In both cases:
- The first step is to place the compass on the given point and draw an arc in a way to intersect with the given segment twice.
- Then, place the compass onto the intersection points respectively and open the compass larger than the previous radius to draw the arcs.
- Draw the line connecting the intersection point of the arcs and the original point. The newly constructed line is perpendicular to the original line.
There are several ways to construct parallel lines. One of the most intuitive ones is using the properties of quadrilaterals with parallel lines. In a rhombus, the congruent sides can be constructed by drawing the radius of a circle.
- First place the compass on the given point.
- Draw a long arc (almost a full circle) that intersects the original segment twice.
- Then, place the compass onto the first intersection point by keeping the compass width constant draw an arc that intersects with the segment.
- Now, use this intersection point to draw another arc that intersects with the previous one. That is in fact the fourth vertex of the rhombus.
- Finally, connect the original point with the latest constructed point to draw the parallel line.
- By this method, one can construct parallel lines as well as a rhombus.
Copying an angle
Share this Polypad with students ask them to copy the angle BAC. Here is the solution:
- Copying means constructing another angle, say , congruent to . The two angles must have the same measure.
- Place the compass point on A and draw an arc.
- Keep the compass width constant, place the compass point on D and draw another arc.
- Place the compass point on the intersection point of the first arc and ray AB and draw another arc passing through the intersection of the initial arc and the ray AC.
- Again, keeping the compass width the same, place the compass point on the intersection of the arc and ray ED, then draw an arc to mark the intersection of both arcs.
- Draw a ray from D through the intersection point (F) of the two arcs. The angle EDF is congruent to angle BAC.
Ask students to measure the angles with the protractor tool to prove their measures are the same. You can also use the fraction slice of to represent the angle measure.
Bisecting an angle (Constructing the angle bisector)
Share this Polypad with students. Here is the solution:
- Set the compass point on B and draw an arc through BA and BC.
- Label the intersection points of the arc and the segments BA and BC as D and E respectively.
- Place the compass point on E and draw an arc in the interior region of the angle ABC.
- Place the compass point on D and draw another arc by keeping the compass width constant.
- Mark the intersection point of the arcs as F.
- Draw BF with a straight edge. BF is the angle bisector of the angle ABC. In other words;