Simple Constructions in Euclidean Geometry Lesson Plans

for

Math 9 class, British Columbia, Canada,

by Max Sterelyukhin, , 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 achord bisects the
chord.

• Measure of the central angle is equal to twicethe measure of
the inscribed angle subtended bythe same arc.

• Inscribed angles subtended by the same arc arecongruent.

• Tangent to a circle is perpendicular to the radiusat 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 so that cm, cm, .

(2) Draw so that cm, ,.

(3) Draw so that cm, cm, 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 . 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?