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Topic 5:
GEOMETRIC CONSTRUCTION SKILLS
Key Words: perpendicular lines, parallel lines, circumcircle, arcs
By the end of this topic, you should be able to:
i) draw perpendicular and parallel lines.
ii) construct perpendiculars, angle bisectors, mediators and parallel lines.
iii) use compass and a ruler to construct special angles (600, 450).
iv) describe a locus.
v) relate parallel lines, perpendicular bisector, angle bisector, straight line and a circle as loci.
vi) draw polygons.
vii) measure lines and angles.
viii) construct geometrical figures such as triangle, square, rectangle, rhombus, parallelogram.
Introduction
In this topic you will learn how to construct lines, angles and geometric
figures. Skills developed from this topic can be applied in day-to-day life.
Sub-topic 5.1: Draw perpendicular and parallel lines
Activity 5.1: Drawing perpendicular and parallel lines
(a) In your groups, list objects in real-life situations that can be used to draw lines.
(b) Use the objects in (a) above to draw perpendicular lines, parallel lines and intersecting lines.
Activity 5.2: Identifying lines
In your groups, take a sheet of paper; divide it into half, then into half
again in the same way. Now fold your paper again. What kind of lines
do you see?
Next, fold the same paper into half in the opposite direction. Unfold
your paper now.
How is the new line you have created, related to the previous lines?
In real-life situations, where do we come across perpendicular lines
and parallel lines?
Which letters in the alphabet have the above lines?
In this sub-topic, you will have more hands-on work on perpendicular
and parallel lines
Sub-topic 5.2: Construction of Perpendicular Lines
Activity 5.3: Construction of perpendicular line from an external point to a given line
In your groups, work in pairs.
Given line segment AB and point C outside the line, construct a perpendicular line from point C to line AB.
Taking the centre as C and any radius, draw two arcs on line AB at x and y.
Now taking x as the centre and any radius, draw an arc below or above the line opposite point C without changing the radius. Taking y as the centre, draw an arc to intersect the previous arc. Join the intersection of the arcs to point C .Compare your answers and make notes.
Activity 5.4: Construction of a Perpendicular line to a given point on a given line segment
In your groups, work in pairs.
Given line PQ and point Z on PQ. Taking Z as the centre and any radius, draw two arcs on either side of Z name the arcs x and y . Now taking x as the centre and any radius draw an arc either above or below the line, without changing the radius now taking y as the centre draw an arc to meet the previous arc join the intersection of the arcs to point Z . Compare your answers with other group members.
Activity 5.5: Construction of a Perpendicular Bisector
In your groups work as an individual.
Given line segment AB. Taking A as centre and AB as the radius, draw two arcs below and above the line, then now taking B as the centre and without changing the radius, draw arcs to meet the previous arcs. Join the intersection of the arc. What do you notice? Compare your work with your group members.
Activity 5.6: Construction of parallel lines
In your groups, work in pairs.
Given line AB and point C outside the line. Take C as the centre, draw an arc at point A taking AB as radius and A as the centre, draw an arc at point B. Now take radius AC and taking B as the centre, draw an arc above B, then taking radius AB and C as the centre, draw an arc to meet the previous arc at D. Join the intersection of the arcs (D) to point C. What do you notice. Name and describe shape ABCD. Compare your answer with members of the your group.
Sub –topic 5.2: Using a Ruler, Pencil and Pair of Compasses, Construct Special Angles
Activity5.7: Construction of special angles
In pairs, construct the following angles: 15°, 30°, 45°, 60°, 75°, 90°, 120°.
In your groups, compare your answers.
Using a protractor, measure your angles.
Sub-topic 5.3: Describing Locus Question
What is the path traced out by the tip of the seconds-hand of a clock in the course of each minute?
Activity 5.8: Discovering what Locus is
In your groups, discuss what happens if a goat is tied to a rope of length 4 metres and around the place where the goat is, there are gardens at a distance of 5 metres.
In your groups, draw sketches of the area where the goat can feed from.
In real-life situations, where are such scenarios applied?
Activity 5.8: Sketching and Describing Loci
In your groups, sketch and describe what happens about the following:
a) A mark on the floor as the door opens and closes.
b) The centre of a bicycle wheel as the bicycle travels along a straight line.
c) A man is walking and keeping the same distance from two trees P and Q.
d) A student is walking in the assembly hall keeping the same distance from two opposite walls.
e) Compare your answers with other groups.
5.3.1: Relating Lines and Angles to Loci
According to the activities above, Locus is a trace of a point under some conditions.
Activity 5.9: Demonstration of some simple Loci
a) In your groups, demonstrate how one can walk the same distance from a given point.
b) How one can walk the same distance from two fixed points.
c) How one can walk the same distance from a line.
d) How one can walk the same distance from two intersecting lines.
In your different groups compare your answers.
Exercise
1. Construct the locus of a point equidistant from a fixed point.
2. Construct a locus of a point equidistant from a given line.
3. Construct the locus of a point equidistant from two intersecting lines.
4. Construct a triangle ABC where AB = 12cm, AC=9cm and Angle BAC = 60°. Find the point with 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. (vi, vii are implied.)
Sub-topic 5.4: Construction of Geometric Figure
Construction of geometric figures most of the time is application of locus.
Activity 5.10: Construction of geometrical figures
In pairs, 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?
In your groups, construct an equilateral triangle with length 6cm.
Construct a circumcircle of the triangle. What type of locus is applied here?
Exercise
1. Construct a triangle ABC in which AB = 8.5, BC = 6cm and angle B = 30°.
Construct a circle through the vertices of the triangle. Work out the area of the circle.
2. Construct triangle PQR with PQ = QR= 7cm angle Q = 45°. Construct a circumcircle of the triangle.
3. Construct a parallelogram ABCD in which AB=5cm, BC=4cm and angle B is 120°.
4. Construct an equilateral triangle ABC of sides 7cm. Bisect AB and BC
and let the bisectors intersect at X. With X as the centre and radius XA, draw a circle.
Situation of Integration
In a village, there is an old man who wants to construct a rectangular
small house of wattle and mud.
Support: A string, sticks, panga, tape measure and human resource.
Resources: Knowledge of horizontal and vertical lines i.e. rows and columns, knowledge of construction of geometric figures.
Task: The community asks you to accurately construct the foundation plan for this old man’s house.
Explain to the class how you have accurately constructed the foundation plan. Discuss whether there are other ways of constructing an accurate foundation plan.
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