Topic 5: Geometric Construction Skills

By the end of this topic, you will be able to:

• draw parallel and perpendicular lines
• construct perpendiculars, angle bisectors, mediators and parallel lines
• use a pair of compass and a ruler to construct special angles of 60° and 45°
• describe a locus
• relate parallel lines perpendicular bisector, angle bisector, straight line and a circle as loci
• draw polygons
• measure lengths and angles
• construct geometric figures such as triangle, square, rectangle, rhombus, parallelogram, etc.
• Geometric Construction

Keywords

• angle
• bisector
• construction
• drawing
• geometric figure
• locus
• parallel lines
• perpendicular lines.
• polygon
• Geometric

Introduction

How do carpenters, engineers, designers, and builders come up with the beautiful and accurate measurements and designs? It all starts with using a set of geometric instruments and skills to come up with lines, angles and shapes to form beautiful designs. In this topic, you will learn how to use angle properties of lines and shapes to solve problems.

Activity 5.1 Drawing perpendicular and parallel lines (work in groups)

What you need: rulers, pencil, notebook, and table.What to do: A B

1. Label the top face of the book as ABCD and top face of the table as PQRS.

2. Identify and draw lines on the sides that cannot meet.

3. Identify and draw lines on the sides that meet at 90°.

4. List other objects in real life that can be used to draw parallel and perpendicular lines.

Parallel Lines

Activity 5.2 Drawing parallel lines using set instruments (work in groups)

What you need: geometry set, notebook.

Precaution: Be careful with sharp geometric instruments not to injure yourself.

What to do: Drawing a line parallel to line KY through point P.

1. Place the 60° set square along line KY and place a ruler along the set square.

2. Holding the ruler firmly, slide the set square along the ruler to touch point P and draw a line through point P using the set square.

3. Repeat the above steps to draw parallel lines.

Perpendicular Lines.

Activity 5.3 Drawing Perpendicular lines (work in groups)

What you need: Geometry set and notebook

Precaution: Be careful with sharp geometric instruments not to injure yourself.

What to do: Drawing a line perpendicular to line AB below at point P

1. Place the set square at point P such that it’s at a right angle

2. Draw a line using the set square.

Constructing Perpendiculars, Angle Bisectors, Mediators and Parallel Lines

In this section, you will construct perpendiculars, angle bisectors, mediators and parallel lines.

Constructing Perpendiculars

Activity 5.4 Constructing perpendicular lines (work in groups)

What yo

What to do:

1. Draw a straight line.

2. Put your compass at any point around the middle of the line and draw two arcs on either sides of the line as shown.

3. Place the compass on one of the arcs on the line and draw arcs above and below the line.

4. Without adjusting the compass, place the campus pin on the other arc on the line to draw an arc above and below the line such that they cross the first two arcs.

5. Use a ruler or straight edge to join the points where the two arcs intersect.

6. Measure the angle between the two lines.

7. Confirm whether the line constructed is perpendicular to the original line.

Angle Bisectors

Activity 5.5 Constructing an angle bisector (work in groups)

What you need: Geometry set, pencil, notebook/paper.

What to do:

1. Draw an angle of any size.

2. Place the compass pin on the vertex of the angle and draw an arc on both arms.

3. From where the arc crosses the arm, make an arc in the angle’s interior.

4. Without changing the compass, repeat for the other arm.

5. Display your work on the wall and move around studying other groups’ work.

Constructing Mediators

A mediator bisects a side of closed plane figure. In the following activity, you will construct mediators on the sides of a triangle.

Activity 5.6 Constructing a mediator (work in groups)

What you need: geometry set, pencil, notebook or paper.

Precaution: Be careful with sharp geometric instruments not to injure yourself.

What to do:

1. Draw a triangle with sides ABC.

2. With the pencil fixed in your compass, place the compass pin at point A and draw an arc above and below line AB.

3. Without adjusting the compass, move it to point B and draw another set of arcs to intersect with the first arc.

4. Draw a straight line joining the intersection of the arcs.

5. Repeat steps 2 to 4 to construct mediators for sides AC and BC. 4.

Exercise 5.4

Draw mediators on each of the sides of a regular pentagon.

Constructing Parallel

Activity 5.7 Constructing parallel lines (work in groups)

What you need: geometry set.

Precaution: Be careful with sharp geometric instruments not to injure yourself.

What to do:

1. Draw a straight line AB

Constructing of Special Angles (60° and 45°)

Constructing an Angle of 60°

Activity 5.8 Constructing an angle of 60° (work in groups)

What you need: geometry set, notebook.

What to do:

1. Draw a straight line using a ruler or straight edge. Place your compass at any point on the line and draw an arc on the line and above the line without changing the compass width.

Construct an Angle of 45°

Activity 5.9 Constructing an angle of 45° (work in groups)

Exercise 5.5

1. Construct angles 30°, 15°, 22.5°, 75°, 135°, using the same procedures.

2. Using a ruler, pencil and pair of compasses only, construct the angles 90°, 180°, 105°, and 300°.

Describing a Locus

When you tether an animal, there is a furthest distance it can reach around the point at which it is tethered. The furthest distance is a circular path. Therefore, the locus of points which are at the same distance from the centre is a circle. A locus is a set of points which satisfies a given condition. When they are many; we call them loci.

Activity 5.10 Locus (work in groups)

What you need: ropes, metre ruler, stick, paper.

What to do:

1. a) Get outside of your class in the school compound.

b) Fix two ropes of the same length between two fixed points A and B.

c) Hold the strings at the centre length and stretch them to opposite directions as shown.

d) Mark the points Q and P and remove the ropes

e) Join point Q to P to form a locus. What is the name of the locus formed.

2. a) Get a rope with length slightly bigger than Length AB as shown

b) Use a stick to stretch the rope at a point and move the stick round to form a locus in which the sum of the distance from two fixed points is constant.

Exercise 5.6

1. Draw the following loci.

a) Parallel lines 2 cm apart.

b) A perpendicular bisector of a line AB 4 cm long.

Relating Parallel Lines, Perpendicular Bisector, Angle Bisector, Straight Lin and Circle as Loci

Activity 5.11 Relating parallel lines, perpendicular bisector, angle bisector, straight line and circle loci (work in groups)

What you need: pairs of compasses, ruler, pencil, paper.

Precaution: Be careful with sharp geometric instruments not to injure yourself What to do:

Work in groups,

1. Construct a circle of any radius.

2. Draw a horizontal line through the centre of the circle

3. Name that line ON.

4. Construct a perpendicular bisector through the centre of the circle perpendicular to ON. Name that perpendicular line KL.

5. Construct the parallel lines below ON and above ON. Name the lines PQ and RM.

6. Measure the distance between ON and the lines PQ and RM.

7. Join O to K.

8. Bisect the angle KON such that the bisector meets KL at W.

9. Measure and state the distance KW and LW.

10. Name the figure KOW formed.

Parallel lines, perpendicular bisectors, angle bisectors straight lines and a circles are all examples of loci.

Exercise 5.7

Follow the above steps and relate parallel lines, perpendicular bisector, angle bisector, straight line and a circle as loci. Use the radius of a circle as: i) r = 3 cm ii) 4.9 cm

Drawing Polygons

Activity 5.12 Drawing polygons (work in groups)

What you need: papers, rulers, cutters, pencils.

Precaution: Be careful not to injure yourself with a cutter.

What to do:

1. Draw and name different polygons on a piece of paper. Make cut-outs of those polygons.

2. Count the number of sides on each cut-out.

3. Measure the lengths and angle of each polygon.

4. Present your work to the whole class.

Exercise 5.8

1. a) Identify any 5 other examples of polygons and draw them.

b) Measure the lengths and angles of each polygon.

2. Measure the length and angles of the polygons below.

3. What do you notice on polygons (a), (b) and (c).

Constructing Geometrical Figures

Geometry is all about shapes and their properties. When you look at a house roof, which shapes can you identify?

A triangle

Activity 5.13 Constructing a triangle

What you need: banana fibres, manila paper, razor blade or cutter, pencil, ruler, markers.

Precaution: Do not play with the razorblade.

What to do: Make a triangular cut out from manila and use it to make a poster indicating children crossing.

Measure the three sides and angles of the poster. Identify any shape in the school with triangular shape.

Example

Exercise 5.9

1. Below are sketches of triangles. Using a mathematical set;

a) Construct the accurate triangles

b) Measure and state the length of AC and BC in (i), PR, and QR in (ii),

c) State the length KW and angle KLW in (iii) above.

d) Measure and state the angles at C and R.

A square

There are so many shapes in the environment in square forms. These include tables, cakes, files, books, boxes and many others.

Activity 5.14 Constructing a square What you need: sticks, threads, banana fibres, pieces of cloth.

What to do:

1. Make a four-sided figure with all sides equal, using sticks, threads and banana fibres.

1. Make many squares of different size.

2. State the size of the angles formed in a square?

3. You can construct the square using geometric instruments.

Exercise 5.10

1. Construct the following squares given one side as

a) 2.8 cm b) 4.5 cm

2. Construct squares whose diagonals are;

a) 8 cm

b) 10cm

Constructing a Regular Hexagon

In the environment, we interact and use items that are hexagonal like glasses, cups, trays, tables, chairs and many more. All these items are manufactured and constructed hexagonally.

Activity 5.15 Constructing a regular hexagon

What you need: geometrical instruments.

Precaution: Be careful with sharp geometric instruments not to injure yourself. What to do: Construct a regular hexagon in a circle of radius 4 cm.

1. Fix a pencil in a pair of compasses

2. Measure a length of 4cm on the ruler using the pair of compasses

3. Draw a circle of that radius.

4. Make a point on the circumference. From that point, make equal arcs through the whole circumference.

5. Join the points where the arcs meet the circumference.

6. Measure and state the length of each side.

7. What is the distance around the hexagon?

8. Display your work and walk around comparing with other groups’ work

Exercise 5.11

1. Construct a regular hexagon of length:

a) 3 cm b) 4.2 cm