CHAPTER SEVEN

GEOMETRICAL CONTRUCTIONS

7.1 Objectives

By the end of this chapter, you should able to:

i. identify mathematical instruments and their uses

ii. differentiate between draw and construct in geometry

iii. bisect any given line or angle

iv. construct special angles

v. construct a triangle or any shape

vi. define the locus of a point

7.2 Introduction

This chapter is dedicated to simple construction. Looking at structure and objects around us there is evidence of geometrical constructions and that make mathematics real to life. Laboratory activities involving geometrical constructs are provided for pupils. By the time students have acquired the abilities and mastered the lessons, mathematics will be viewed as a living, non-formula-based topic. In geometry, it is essential that pupils recognize the distinction between draw and build. In geometry, just a ruler and a pair of compasses are used to "build," however in "draw," any tool deemed required for producing such a drawing may be used. In this chapter, we will discuss the building of bisectors of lines and angles, as well as the development of forms and loci.

7.3 Components of Mathematical Instruments

Mathematics instruments are sometimes known as geometric tools. They tools used in constructing different types of geometric shapes. The most instruments (tools) that are used within the field of geometry, including the ruler, divider, protractor, set squares and compass. The description and uses of the tools are discussed in details in next unit.

Ruler Protractor Pencil

Dividers Compass Set Squares

7:4 Uses of Mathematical Instruments

Each of these tools is used in geometric constructions to provide psychomotor activities.

Ruler

It is also known as a straightedge or line gauge. It is a tool utilized in geometry and several engineering applications. A ruler is utilized for generating straight lines and measuring line segment lengths. It may also be used to measure the object's distance. One side is graduated in centimeters and millimeters, while the other side is graduated in inches. The space between the markings on the ruler is known as the Hash.

Compass

It is a V-shaped device that is characterized as a drafting tool with a sharp tip in the center and a pen or pencil at the other end. Adjustable is the distance between the pencil and the pointer. It is used to draw circles, arcs, and angles. In addition to marking equal lengths, it is one of the tools needed to construct a geometric circle.

Protractor

It is a semicircular disc that is used to measure and draw angles. It is graded from 0 to 180 degrees and can measure any angle within its range. 0 to 180 degrees are marked from left to right and vice versa. The protractor's inner and outer readings complement one another, and the total of the two readings equals 180 degrees.

When measuring an angle using a protractor, the following procedures must be observed:

§ if the angle to be measured is on the left side of the protractor, the outer reading is used; if the angle to be measured is on the right side of the protractor, the inner reading is used.

Divider

Divider resembles compasses in appearance because to its "V"-shaped construction. However, the "V" has pointers on both ends. The adjustable distance between them is used to measure and compare lengths. Frequently, the divider and compass are used to denote distance or division.

Set-Squares

The set-square is one of the mathematical tools typically seen in geometry boxes. These are the triangle plastic pieces with the space between them eliminated. There are two possible sets of set-squares. One angle is 45 degrees, while the other ranges from 30 to 60 degrees. The 45-degree set square has a 90-degree angle, while the 30-60-degree set square has a straight angle. The 45-degree set square is used to draw vertical lines, as well as parallel lines, perpendicular lines, and standard angles, among other things.

Student Activity 1

Draw out all instruments in mathematical set and their uses

7.5 Teaching of Construction

You must note the following when carry out construction. These are:

1. A very sharp pencil.

2. A very straight ruler.

3. A clear eraser.

4. A very good pair of compasses, so as to enable you arrives at a correct and neat construction.

1. To bisect a given line

To bisect a given straight line, a ruler and a set of compasses are required. In order to do the bisection, the following procedures must be taken

(a) Draw the given line AB

(b) With any convenient radius greater than half AB and center A, construct an arc above and below, the line

(d) The line drawn through the points of intersections of the arcs will cut line AB at E

(e) Hence |AE|=|EB|

2. To bisect a given angle

In constructing a line called a bisector of an angle, you required a pair of compasses, a ruler, and a sharpened pencil. The following procedures are followed.

(a) Given angle XYZ

(b) Using Y as center, and any radius draw an arc to cut YZ at A and YX at B ii

(c) With center A and B and equal radii [ either equal to AB or greater than half of AB], draw the area to meet at T. Join T

(d) Therefore XŶT= TŶZ.

3. Construction of a perpendicular to a line

In carrying out this kind of construction, there are two possibilities. The perpendicular can be from.

(a) A point X to the line

(b) A point X outside the line

The same construction instruments are needed as those in the preceding constructions.

(a) A perpendicular from a point X on the line.

Let the line be AB. With X as the center and a reasonable radius, draw an arc to cut AB at M and N. Then with M and N as the center and same radius, draw arcs to meet at Y. Then join XY, XY is the required perpendicular from the point XY to AB.

(b) A perpendicular from a point X outside the line.

Let the line be AB with X as centre and any convenient (or reasonable) radius, make an arc cutting the line AB at P and Q. then, with P and Q as centre and the same radius, make arc cutting each other at Y. then join XY. XY is the required perpendicular to the line AB

4. Construction of an angle of 90⁰

Construction of angle of 90⁰ is the same as a perpendicular line to a given line at a given point.

(a) Draw line AB\

(b) Mark any point C on AB

(d) With center D and E at different times and same radius draw arcs to intersect one another at point F

(e) Join CF. note: FCB=FCA=90⁰

5. Construction of angle of 45⁰

(a) Construct angle BAC= 90⁰

(b) Bisect angle 90^o to give angle 45⁰

6. Construction of an angle of 60⁰

(a) Draw a line AZ

(b) Mark a point X on AZ

(d) With center D and the same radius as XD, draw an arc to cut the previous one at E

(e) Draw a line from X through E [line XY]. YXZ is the required angle 60⁰

To construct an angle of 30⁰

(a) Construct an angle of 60⁰ (as indicated above)

(b) Bisect the angle of 60⁰

7. Construction of an angle of 15⁰

(a) Construct the 60⁰angle

(b) Bisect the angle 60⁰ to give 30⁰ angle [as shown above]

(d) ABC is the required 15⁰ angle

8. Construction of multiple angles

Construction of angles like 60⁰, 120⁰ and 90⁰ are directly constructed without any difficulties. But some special angles are done through adding or subtracting as the case may be. Such special angles are

(a) Construction of angle 105⁰

Angle 105⁰ is the sum of angle 90⁰ and 15⁰ie 90⁰+15⁰=105⁰.hence, angle 105⁰ is constructed.

(b) Construction of angle 75⁰

Angle 75⁰ is the sum of angle 60⁰an 15⁰

(c) Construction of angle 150⁰

Angle 150⁰ is the sum of angle 120⁰ and 30⁰, or substruct 30⁰ from 180⁰

(d) Construction of angle 135⁰

Angle 135⁰ is the sum of 90⁰ and 45⁰ or 120⁰ and 15⁰.

Student Activity

1. Construct angle 60⁰, 120⁰ and 90⁰

2. Draw a line AB, locate a point X on the line

With point X construct angle 135⁰. Bisect the just constructed angle.

3. Draw a line |XY|=10cm. locate a point P such that |XP|=2cm locate another point Q on the line such that PQ=5cm. construct the perpendicular bisector of PQ.

9. Draw a parallel line.

(a) Draw the line XY. Make points ‘a’ and ‘c’ near to the ends of the line respectively and ‘b’ about the center of the line.

(b) With centre’s a b and c and a radius equal to the distance away from the required parallel, draw arcs.

10. Construction of a parallel line to a given straight line through a point

(a) Draw straight line XY and a point P which is not on the line

A parallel line can be constructed to line XY through point P thus:

(b) Mark any two points A and B on XY. With centre P radius equal to AB, draw an arcs S, with centre B and radius equal to AP. Draw another is to cut S at T.

11. Construction of an Equilateral Triangle.

This is based on the construction of an angle of 60⁰ but the lines are joined to form the required triangle. An equilateral triangle is a triangle with all angles equal.

12. Construction of an Isosceles Triangle

An isosceles triangle is a D with two of its three sides equal and the two angles facing the equal side, equal too. Now let us look at how it illustrated construction.

(a) Draw line AB

(b) With A as centre and any reasonable radius not equal to AB, make an area of a circle.

(d) Join AC and CB. Hence, DABC is an isosceles D with AC=BC

13. Simple Constructions with Circle & Triangle

There are three simple constructions, which deal with triangles and circles. These constructions are.

In doing the construction suppose you have Triangle ABC on which a circum-circle is to be constructed:

a. Choose any two sides, say AB and AC

b. Construct the perpendicular bisector of the two sides AB and to meet at M.

c. Using M as the centre and radius equal to MB, draw a circle. The circle will pass through vertices A, B; C. this is the required circum-circle.

B. The inscribed circle [in circle] is a circle that is constructed inside an already triangle. The circle will touch the sides if the triangle sees the figure below:

The construction of an in-circle is done as follows:

Given a triangle XYZ to which an in circle is to be constructed

i. Choose any two angles of the triangle

ii. Construct the bisector of the two angles to meet at any point P.

iii. Construct a perpendicular from P to A on XY

iv. Then with centre P and radius PA, draw a circle.

The circle is the required in circle (in cribed circle)

The in circle is also called inscribed circle.

C. The E-scribed circle of a triangle ABC

It is the circle outside the triangle as shown in figure below with the three sides of the triangle as tangents to the circle. The circle is exterior to the triangle and called the ex-scribed circle.

14. Locus of a point.

The locus of a point is the path traced by that point under specific conditions. All point on that path satisfies the condition (S). The plural of locus is loci

Example of same common 2-dimensional loci

(i) The locus of a point which moves with a constant distance from a fixed point in a circle; with the fixed point as the centre and the constant distance as the radius. This is illustrated as below

(ii) If the point can move anywhere in space with the constant distance from a fixed-point O, then the locus is a sphere with O as the centre, and r, the constant distance as the radius of the sphere.

(iii) The locus of a point in a plane, which is always equidistant from fixed points is the perpendicular bisector of the line joining the two points see the figure below

Student Activity 2

1. Distinguish between construction and drawing

2. a. what is the locus of a point

a. Sketch the locus of a point in space moving with a constant distance from that fixed point. Name the product.

3. Construct a triangle PQR such that |PQ|= 7cm, |PR|= 6cm and |QR|= 8cm. construct the bisector of P meeting QR at X. Construct the bisector of PR meeting PX at Y. Measure |QY|

4. Describe how you would construct angle 120⁰

7.6 Summary

In this chapter, you have learnt about the following.

· Mathematical instruments and their uses

· Differences between drawing and construction in geometry

· Construction of perpendicular line from a point X to the line and a point X outside the line

· Bisection of an angle

· Constructions of special angles 30⁰, 60⁰, 45⁰, 90⁰

· Construction of triangles such as equilateral, scalene, etc

· Construction of parallel lines

· Locus of a point

References

National Teachers’ Institute, Kaduna (NCE/DLS) Mathematics 2

National Teachers’ Institute & National Open University of Nigeria: PED 243 Measurements and Shapes