What is a construction?
Circle Constructions Task
In geometry, there is a difference between drawing a diagram and constructing a diagram. In constructions, your tools
are limited to a pencil, a compass, and a straightedge (a ruler with no measurements), while a diagram can be drawn
with anything. Additionally, the steps for drawing a construction must be justifiable using geometric theorems or
axioms. The main ones are summarized from Hilbert’s axioms below:
1) There exists at least one point on a plane, segment, or line. (from undefined terms)
2) There exists only one line (or segment) through any two points on a plane. (from postulates 1-1 & 1-2)
3) A circle with a radius congruent to a given segment creates congruent segments. (from postulate 3-1)
I have made a few tutorials for these basic constructions and a couple more that you might need for this task listed
below. You can use either one of the digital apps to construct the necessary items or use a compass and straightedge.
(Geogebra / Robo Compass) Robocompass relies a bit more on the coordinate plane to place points than Geogebra does,
but it lets you construct first and animate later.
Basic Constructions Demos:
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Copying a line segment
Copying an angle
Bisecting a line segment
Bisecting an angle
(Geogebra
(Geogebra
(Geogebra
(Geogebra
/
/
/
/
Robo
Robo
Robo
Robo
Compass)
Compass)
Compass)
Compass)
Directions
If you work with a group (of no more than 3), then each member of the group will pick a different construction from
each section so that all constructions are covered. Your grades will not be dependent on each other.
Choose
★
★
★
one of these:
Inscribe a circle in a triangle (Start with a triangle)
Circumscribe a circle around a triangle (Start with a triangle)
Construct a tangent line from a point outside a circle (Start with a circle and an external point)
AND, one of these:
★ Inscribe an equilateral triangle (Start with a circle)
★ Inscribe a square (Start with a circle)
★ Inscribe a regular hexagon (Start with a circle)
For EACH construction…
Create a screencast video of each construction you selected. Add the links to this document.
Constructions
#1→
#2→
Links
List the steps you took to complete your constructions and identify which axiom or property allowed you to
take that step. (see example below for the tutorials-- this is similar to a proof). If you summarize the steps
for copying and/or bisecting, then you can reference the tutorials instead of the axioms/properties of each one.
Copying a Segment
Steps
1.
Justifications
Draw a line segment AB and a point C not on AB.
1.
Axioms 1 & 2 (“Given”)
2. Draw a circle centered at C with a radius equal to AB.
2. Axiom 2
3. Mark a point D on circle C
3. Axiom 1
4. Draw segment CD
4. Axiom 2
Copying an Angle
Steps
1.
Justifications
Draw an angle BAC and a ray/segment DE not
intersecting angle BAC.
1.
Axioms 1 & 2
2. Mark a point F on segment AC.
2. Axiom 1
3. Draw a circle centered at A with a radius equal to AF.
3. Axiom 3
4. Mark the intersection between circle A and segment AB
and label it point G.
4. Axiom 1
5. Draw a circle centered at E with a radius equal to AF.
5. Axiom 3
6. Mark the intersection between circle E and segment DE
and label it point H.
6. Axiom 1
7. Draw a circle centered at H with a radius equal to FG.
7. Axiom 3
8. Mark the intersection between circle H and circle E and
label it point I.
8. Axiom 1
9. Draw a segment/ray from E through I to create angle
DEI.
9. Axiom 2
Bisecting a Segment
Steps
1.
Justifications
Draw a segment AB
2. Mark a point C over halfway between A and B
1.
Axiom 2
2. Axiom 1
3. Draw a circle centered at A with a radius equal to AC
3. Axiom 3
4. Draw a circle centered at B with a radius equal to AC
4. Axiom 3
5. Mark the intersections between circles A and B and label
them D and E.
5. Axiom 1
6. Draw a line through points D and E.
6. Axiom 2
Bisecting an Angle
Steps
1.
Justifications
Draw an angle BAC
1.
Axioms 1 & 2
2. Mark a point D on segment AB
2. Axiom 1
3. Draw a circle centered at A with a radius equal to AD.
3. Axiom 3
4. Mark the intersection between circle A and segment AC
and label it as E
4. Axiom 1
5. Draw a circle centered at E with a radius equal to DE
5. Axiom 3
6. Draw a circle centered at D with a radius equal to DE
6. Axiom 3
7. Mark the intersection between circles E and D in the
interior of angle BAC and label it as F
7. Axiom 1
8. Draw a line through points A and F to bisect the angle
8. Axiom 2