CHAPTER FIVE 

Chapter 5: GEOMETRIC CONSTRUCTION

SKILLS

Learning objectives

By the end of this topic, the learners should be able to

• Draw perpendicular and parallel lines

• Construct perpendiculars, angle bisectors, mediators and parallel lines

• Use a pair of compasses and a ruler to construct special angles

• Describe a locus

• Relate parallel lines,perpendicular bisector ,angle bisector,straight lines and a circle as loci

• Draw polygons

• Measure lengths and angles

• Construct geometrical gures such as triangle,square ,rectangle,rhombus ,parallelogram

In this topic you will learn how to construct lines, angles and geometric gures. Skills developed

from this topic can be applied in day-to-day life.

5.1 Parallel ,Perpendicular and intersecting lines

Identify the lines below

5.2 Construction of Perpendicular Lines

Activity : Construction of perpendicular line from an external point to a given line

Given line segment AB and point P outside the line, construct a perpendicular line from point P to line AB.

STEPS

1. Taking the centre as P and any radius, Place the campus at point P and draw two arcs to cut line AB .

2. Taking A as the centre and any radius, draw an arc below or above the line opposite point P .

3. Without changing the radius and taking B as the centre, draw an arc to intersect the previous arc at point Q.

4. Join the intersection of the arcs from point P to Q

Figure 5.1: Construction of perpendicular line from an external point to a given line

Figure 5.2: Construction of a Perpendicular line to a given point on a given line segment

Activity : Construction of a Perpendicular Bisector

Given line segment AB ,draw a perpendicular bisector

STEPS

1. Draw a line segment with end points A and B .

2. Place the point of compass at A ,stretch out the compass until more than half of the length of AB.

3. Draw an arc on either side of the line segment

4. Keeping the radius of the compass constant, place the point of the compass at point B and draw an arc on either sides of the line segment.

5. Join the intersection of the arc at point C and D.

Figure 5.3: Construction of a Perpendicular bisector

5.3 Construction of parallel lines

STEPS

1. Draw a line segment with end points AB and a point C outside the line .

2. Draw an arc at point A taking AB as radius and C as the centre.

3. Taking A as the centre and AB as the radius ,draw an arc at point B

4. Taking AC as the radius and B as the centre ,draw an arc above B.

5. Taking AB as the radius and and C as the centre, draw an arc to meet the previous arc at D.

6. Join the intersection of the arcs at point D to C.

7. The line segment AB is parallel to CD

5.4 Construction of special angles

1. In construction, of angles, we use a ruler, pencil and pair of compasses only.

2. The angle bisector method can be used to create other angles. Thus, an angle of 30° is obtained by bisecting an angle of 60°.

3. The supplementary angle construction method can be used to get obtuse angles. Thus, an angle of 120° is obtained by constructing an angle of 60°.

Activity : Construction of special angles

Using a pencil, ruler and pair of compasses only, construct the following angles:

1. 90° 

2. 60° 

5.2 Exercise Set

1. Using a pencil, ruler and pair of compasses only, construct the following angles:

(a) 30° 

(b) 15° 

(c) 7.5° 

(d) 45° 

(e) 75° 

(f) 22½°

(g) 120° 

(h) 135° 

(i) 165° 

(j) 150° 

2. Using a protractor measure the angles constructed above

5.5 Describing a Locus

Locus of a point is the path which it describes as it moves

Activity : Discovering what Locus is

EXAMPLE 1

• What is the path traced out by the tip of the seconds-hand of a clock in the course of each minute?

• The Second hands of a clock moves around the clock and creates a circular path. The tip of each hand is always the same distance (equidistant ) from the centre of the clock. The locus the second hand of a clock create is a circle

Figure 5.4: Path traced by second hand of a Clock

EXAMPLE 2

• Describe what happens if a cow is tied to a rope of length 4 metres and around the place where the cow is, there are gardens at a distance of 4 metres.

• The cow rotates around creating a circular path. Therefore the locus at a distance of 4 metres from the centre (the stake), is a circle with a center and a radius, of 4 metres.

Figure 5.5: Path traced by a cow

Activity : Sketching and Describing Loci

Sketch and describe what happens about the following:

1. A mark on the floor as the door opens and closes.

2. The centre of a bicycle wheel as the bicycle travels along a straight line.

3. A man is walking and keeping the same distance from two trees P and Q.

4. A student is walking in the class keeping the same distance from two opposite walls.

5.6 Relating Lines and Angles to Loci

Locus is a set of points which satisfies a certain condition.

Activity : Demonstration of some simple Loci

1. Demonstrate how one can walk the same distance from a given point.

2. How one can walk the same distance from two fixed points.

3. How one can walk the same distance from a line.

4. How one can walk the same distance from two intersecting lines

5.7 Construction of Loci

Activity : Construction of a locus at a point equidistant from a fixed point.

The locus of points that are at a constant distance from a fixed point is a circle with radius equals to constant distance.

EXAMPLE

Construct the locus of a point Q at a constant distance of 2 cm from a fixed point P.

Activity : Construction of a locus of a point equidistant from a given line

The locus of points that are at a constant distance from a straight line is a pair of parallel lines at a constant distance from the given straight line.

EXAMPLE

Construct the locus of a point P that moves a constant distance of 2 cm from a straight line AB

NOTE

• The locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius i.e. Given a fixed point, the locus of points is a circle.

• The locus of the points at a fixed distance, d, from a line with end points AB, is a pair of parallel lines at a distance, d(apart) from AB and on either side of AB i.e. Given a straight line, the locus of points is two parallel lines.

• The locus of points equidistant from two given points, A and B, is the perpendicular bisector of the line segment that joins the two points..i.e Given two points, the locus of points is a straight line midway between the two points.

• The locus of points equidistant from two intersecting lines, L1 and L2, is a pair of bisectors that bisect the angles formed by line L1 and L2.i.e Given two intersecting lines, the locus of points is a pair of lines that cut the intersecting lines in half.

5.3 Exercise Set

1. Construct the locus of a point Q that moves a constant distance of 3 cm from a straight line XY

2. A dog is on a lead tethered to a post in the corner of a garden. The lead is 5 cm long.

Describe the locus of the dog with a sketch.

3. Construct the locus of a point equidistant from two intersecting lines.

5.8 Construction of Geometric Figure

Construction of figures is an application of the locus, since during inscribing and circumscribing we use the knowledge of angle bisector.

Activity : Construction of geometrical figures

Steps for circumscribing a circle on a triangle. .

• Construct the perpendicular bisector of one side of triangle.

• Construct the perpendicular bisector of another side .

• Where they cross is the center of the Circumscribed circle.

• Place the compass on the center point, adjust its length to reach any vertex of the

triangle, and draw your Circumscribed circle

EXAMPLE

Using a pair of compasses ,ruler and pencil only, construct a triangle ABC in which AB=5cm, ‹BAC=70° and ‹ABC =500

1. Measure and record the lengths BC and AC

2. Construct a perpendicular bisector of the line segments BC and AC

3. Using the meeting point of the perpendicular bisectors as your center,draw a circle to pass

through the vertices of the triangle

4. Measure and record the radius of the circle

5. Calculate the area of the circle

SOLUTION

2. BC= 6.1cm and AC=5.6cm

3. Radius=3.2cm

Steps for inscribing a circle in a triangle.

• Bisect two angles of a triangle

• The angle bisector will intersect at the incenter(centre point)

• Construct a perpendicular from the centre point to one side of the triangle..

• Place the compass at the centre point and adjust its length up to where the perpendicular crosses the triangle, and then draw the inscribed circle.

EXAMPLE

2. Construct a perpendicular from the centre point to one side of the triangle

3. Measure and record the radius of the circle

4. Calculate the area of the circle

5.4 Exercise Set

1. Using a ruler, pencil and compass ,construct a triangle ABC where AB = 7cm, AC=5cm, ‹BAC = 60° . Find the point with in the triangle where the distance from that point to all the vertices of the triangle is equal. Taking that point as the centre and the distance from the centre to the vertices as the radius draw a circle.

HINT:

We need to construct a circle inscribed in triangle and this can be done by making

angle bisector of two sides, the point where it intersect will be incentre. (The angle

bisector is the locus where points are equidistant from two sides)

2. Construct a perpendicular bisector of any line segment. Measure the distance from the perpendicular line to any of the points on either side of the perpendicular bisector. What have you found out? Construct an equilateral triangle with length 6cm. Construct a circumcircle of the triangle. What type of locus is applied here?