Simple Constructions in Euclidean Geometry Lesson
Plans
for
Math 9 class, British Columbia, Canada,
by Max Sterelyukhin, msterely@sfu.ca, April 2009
Introduction
- This topic is to be done in two lessons, with introduction to constructions
and some extensions.
- Expected preliminaries that students are required to have are simple circle
geometry results learned in grade 9 circular geometry section.
- PLO’s:
Solve problems and justify the solution strategy using circle
properties including:
• Perpendicular from the centre of a circle to a chord bisects the
chord.
• Measure of the central angle is equal to twice the measure of
the inscribed angle subtended by the same arc.
• Inscribed angles subtended by the same arc are congruent.
• Tangent to a circle is perpendicular to the radius at the point of
tangency.
- Motivations for inclusion of constructions:
 Provides an extension to circular geometry and its applications.
 Provides more exposure to proof in geometry.
 Provides more exposure on hands-on activities (working with rulers
and compasses, as well as software).
 Great topic to be taught by inquiry.
Lesson 1
Objectives
- Students will develop an overview of constructions in geometry in general.
- Students will be exposed to elementary constructions in geometry.
Try this together
(Students are divided into 6 groups, 2 groups get the same task; after some
time they work on these, each task is discussed)
(1) Draw ABC so that AB  36 cm, AC  2.7 cm, A  40  .
(2) Draw ABC so that AB  4 cm, A  62  , C  54  .
(3) Draw ABC so that AB  5 cm, BC  4 cm, AC  6 cm.
Which tools did you need to use and how?
Exploration 1
(Students and teacher discuss together)
With the aid of compass and ruler without scales construct a line segment
equal to the given one:
Definition
(Written on the overhead or board, students copy)
Constructions in geometry are such questions in which a solution is
obtained by constructing a geometric figure with the aid of compass and
ruler without scales.
Exploration 2
(Each group is given a construction to be tried, then two group
representatives come up to the front to present what was done)
(1) On the given ray construct a line segment equal to the given one.
(2) On the given ray construct an angle equal to the given one.
(3) Given a point on the line, construct a perpendicular line through this
point.
(4) Construct a bisector of the given angle.
(5) Construct a midpoint of the given line segment.
(6) Given a line and a point (not on the line), construct a line through this
point perpendicular to the given line.
Structure (ACPI):
(Written on the overhead or board, students copy)
Each construction is broken down into these 4 steps:
(1) Analysis: rough sketch of construction.
(2) Construction according to plan in (1).
(3) Proof to show that the result is what’s required.
(4) Inquiry: determine when a construction has a solution and when not.
Homework
(Written on the overhead or board ahead of time, students copy)
Suppose we have ABC . Construct:
(1) Angle bisector AK.
(2) Median BM.
(3) Height CH.
Lesson 2
Objectives
- Students will be exposed to further constructions in Euclidean geometry.
Homework Check
(Student per number (1)-(3) to the board to show the solution)
Demonstration
(Using Geometer’s sketchpad, present solution to the following problem)
Given a circle and point A, not on the circle, as we as line segment PQ,
construct a point M on the circle so that AM=PQ. Does this question always
have a solution?
Explorations
(Students are partitioned into 6 groups, each group gets a task to complete,
two representatives from each group to show solution on the board when
done, students are welcome to use sketchpad)
(1) Construct an angle of: (a) 45
(b) 22 30’
(2) Given any triangle ABC, construct intersection point of angle bisector
AL and height BD.
(3) On the given ray construct angle equal to quarter of the measure of the
given angle.
(4) On the given ray construct angle 1.5 times of the measure of the given
angle.
(5) Construct and angle of 135. Then construct a point equidistant from
arms of this angle.
(6) Given triangle ABC below, construct points X and Y so that XA=XB,
YA=YB.
A
C
B
Homework
How do we divide an angle of 54 into 3 equal angles with the aid of
compass and ruler without scales?