2.1.4 Circles

A circle is formed when a continuous line (circumference) starts at 0_ and ends at 360. It is a set of points that are equidistant from a fixed point called the centre. The radius (r on the diagram below) is the distance from the centre of the circle to any point on the circle and can be shown as a line segment connecting the centre to a point on the circle.

The diameter is a line segment that connects two points on the circle and goes through the centre of the circle. It is always twice as long as the radius. A chord (line segment PQ on the diagram below) is any line segment whose endpoints are any two points on the circle. The circumference of a circle is the distance around the circle.An arc of a circle is the set of all points between and including two given points. One way to measure it is in degrees. Keep in mind that the whole circle is 360 degrees. When naming an arc, it is best to use three points – the two endpoints and a point in between – versus just the two endpoints. The reason is you can go clockwise or counter clockwise, which can make a difference when looking at the length of an arc. Arc ADC would start at point A and go clockwise through D and end at C. Arc ADC is a 95 degree arc. Arc ABC would start at point A and go counter clockwise through B and end at C. Since a circle is 360 degrees, then Arc ABC is a 360 – 95 = 265 degree arc. A line is tangent to a circle if it intersects the circle at exactly one point.

The tangent line and the radius of the circle that has an endpoint at the point

of tangency are perpendicular to each other.

2.2 ANGLES

An angle measures the amount of turn. As the angle increases, the name changes

2.2.1 Right Angles

A right angle is an angle that is formed when 2 lines intersect at 90°.

2.2.2 The Acute Angle

An acute angle is any angle smaller than 90° i.e. smaller than a right angle.

2.2.3 The Straight Angle

The straight angle is an angle that equals 180. Any straight line is 180 and is therefore a straight angle.

2.2.4 The Obtuse Angle

Obtuse angles are any angles greater than 90° but smaller than 180° i.e. it is a rotation between a right angle and a straight angle.

2.2.5 The Revolution

A revolution is any angle that is 360°. (I.e. 2 straight angles or 4 right angles)

2.2.6 The Reflex Angle

A reflex angle is any angle greater than 180° but less than 360° i.e. it is an angle between a straight angle and a revolution.

2.2.7 Supplementary Angles

The sum of all the angles that lay along a straight line, will add up to 180°.These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°.

The angles don’t have to be together. These two are supplementary because 60° + 120° = 180°

2.2.8 Complementary Angles

Two Angles are Complementary if they add up to 90 degrees (a Right Angle). These two angles (40° and 50°) are Complementary Angles, because they add up to 90°.

.2.9 Opposite Angles

When two lines intersect each other, vertically opposite angles are equal:

2.2.10 Parallel and transversal lines

Parallel

Parallel lines create many angles when a line intersects them. PQ//RS line PQ is parallel to RS (// means parallel to) and CD intersects lines PQ and RS. CD is called a transversal CD cuts PQ at A and RS at B to create many angles. Remember previous theorems:

1 = ∠3 and ∠2 = ∠4 (pairs of opposite angles on vertex A)

5 = ∠7 and ∠6 = ∠8 (pairs of opposite angles on vertex B)

∠1 + ∠2 = 180°; ∠3 + ∠4 = 180°; ∠1 + ∠4 = 180°; and ∠2 + ∠3 = 180° (angles along a line on line PQ at vertex A)

5 + ∠6 =180°; ∠7 + ∠8 = 180°; ∠5 + ∠8 = 180° and ∠6 + ∠7 = 180°(angles along a line on line RS at vertex B)

The pairs of angles created by the transversal on the parallel lines can be named as follows:

2.2.11 Alternate Angles

4 and 6; 3 and 5; are called Alternate angles.

The alternate angles are equal to each other in pairs, i.e. 3 = 5 and 4 = 6.

Alternate angles are always inside parallel lines.

In conclusion basic geometry involves many terms which needs to be known in definition and also in application:

2.2.12 Perpendicular Angles

When two lines meet each other at 90° then they are perpendicular to each other.

AB meets CD at 90°

2.2.13 Interior Angle

An Interior Angle is an angle inside a shape.

2.2.14 Interior Angle

An Interior Angle is an angle inside a shape.

Exercise

1. Identify the following triangles.

2.Identify each of the following quadrilaterals

. Draw the following figures with the correct tools.

a) rectangle with measurements 40mm x 50mm
b) square with one side 35mm
c) triangle with sides 25mm x 35mm x 50mm
d) circle with a radius of 5mm
The T-square Tiling Company makes ceramic floor tiles. Note: The square tiles that are shown are the same size.

SR and ZY have the same length.

Justify how you know the shaded areas PSRQ and WZYX are the same area.

Calculate the area of the given trapezoid.

Show your work.

2.3 ESTIMATING, MEASURING AND CALCULATING

Estimation allows us to arrive at a ‘nearly correct’ answer that is close enough for all

practical purposes. Estimation is used when an exact answer is not yet required.

When we want to estimate the size of a room, we will not necessarily measure it with a measuring tape, but give large steps that are roughly equal to a meter. In this case we are ‘estimating’ the size of the room.
However, when we need to work out the exact answer, then we will measure the exact space with a suitable measuring instrument. This way we can establish an exact answer.
We want to lay a carpet in a specific room. We will use a measuring tape to get the exact measurement. Otherwise, the carpet may be too small for the room or even too big.
No matter whether you have estimated or measured, if you want to work out, for instance, the size of a room, you will need to calculate to arrive at a suitable answer.

2.3.1 Length and breadth

Length is always the longer side(s) of a shape, while breadth is the shorter sides of a shape. In the example below, length is indicated by the double line while breadth is indicated by the single line. Length and breadth is measured in meters. Any shape that has length and breadth is a two dimensional shape. You can estimate the length and breadth with your fingers. The length of one side is about six fingers long (i.e. one finger may be _2cm wide, so we can estimate that the length of this is 6 x 2 = 12cm) and the breadth is about 2 fingers long i.e. 2 x 2 =4cm. Eventually you will be able to estimate the lengths by merely measuring it with your eye.

2.3.2 Perimeter and Circumference

Perimeter

Perimeter is the distances from one point on the outside border of a shape, all the way around, back to the same point again. Perimeter and circumference is measured in meters.

To calculate the perimeter of a rectangle, square, parallelogram:

P = 2 length + 2 breadths and the answer is in mm, cm, m or km

To calculate the perimeter of any straight lined shape with more than four angles:

P = total sum of the length of all straight sides.

2.3.3 Circumference

Circumference is similar to perimeter, but circumference is the word we use to describe the ‘perimeter’ of a circular shape.

To calculate the circumference of a circle:

C = 2πr (Where π is 22/7 or 3, 14)

2.3.4 Area

Area is the amount of space a shape takes up in two dimensions i.e. length and breadth. If you measure a space such as a room, then you will estimate the length an breadth by using a stride. A stride is a very large step and is the distance between the heel of the back foot and the toe of the front foot. Sometimes the area that we have to calculate has an irregular shape. It seems impossible to then calculate the area, but a little clever thinking will help you to identify regular shapes within the irregular shape and you can then apply the regular formula.

Rectangle

To calculate the area of any rectangular shape we use the formula: A = lb (length x breadth) and the answer is in mm2 or cm 2 or m2 or km2

Circle

To calculate the area of a circle: A = πr2

Triangle

To calculate the area of a triangle:

A = 1/2base x height

Trapezium

There are two methods to determine the area of a trapezium. One method is by formula and the second method is by breaking the shape into rectangles and triangles.

A= ½ sum of // sides x perpendicular height

= ½((24 + 8) x 5

= 80m2

Method 2:

A = Area 1 + area 2 + area 3

= ½ x 4 x 5 + 8×5 + ½ x 12×5

=10 + 40 + 30m2

=80m2

2.3.5 Volume

Volume is the space that a container can take on the inside. In order to determine the volume of a container, we need to add another dimension to the shape i.e. the height or depth. Up to now, we have calculated the area of a shape, working with the length and the breadth. The length is one dimension; the breadth is the second dimension. If we now add height or depth to a shape, we add a third dimension and we can see how much go into it.

2.3.6 Angles

Where two lines, that are not parallel, intersect, and angle is formed. We measure angles in degrees (°) and we use a protractor to measure angles. All angles on the lines intersecting below are indicated by a dot.

We use a protractor as follows:

Place the 0° line of the protractor on top of the base line of the angle (in this case ABD).

Make sure the centre point of the protractor lies directly on top of where the two lines meet (point B). Start from the 0° point and move clockwise to where the secondary line cuts through the point on the protractor (in this case 55°).

If you want to measure angle CBD, your baseline would now be BC and you would place the 0° line of the protractor on top of BC, making sure the central point of the protractor is on top of point B and you will take the measurement(∠CBD will be 110° in this case).

There are a number of common angles that we need to familiarize ourselves with. For information on these angles go back to section 1.4

2.4 DRAWING SCALES

Drawing scales are used to reduce the size of a large article so that it can be represented on a piece of paper. Maps normally use scale to indicate to which extent a piece of land has been reduced. The same principle is used in engineering drawings to illustrate the size of a component or engineering object.

A scale is the ratio of

the distance between two points on a map vs.
the actual distance between two points on a surface

Scales can be represented in one of three ways:

as a ratio 1:1 500 000
as a fraction 1/1 500 000
as a graphic scale

Which means that for each 1 unit of measurement on the map, the distance is 1 500 000 of the same units of measurement on the real surface.

A scale drawing is a drawing where the accurate dimensions of an object or figure or area are reduced or enlarged. The scale drawing is the exact duplication of the original figure, shape or object, but it is smaller. An area is shown on a map as the drawing below. The scale on the map is

1:1 500 000. We measure the length and breadth of this drawing, which is 10cm x 5cm. We can then calculate the actual size of the area land, represented on the map.

1 : 1 500 000

1 x 5 : 1 500 000 x 5

5 : 7 500 000

So for every 5cm we measure on the drawing, the real measurement on land is

7 500 000cm (or 7,5 km).

1 : 1 500 000

1 x 10 : 1 500 000 x 10

10 : 15 000 000

The length in this case will be 15km on land.

Scales are very convenient in representing large areas on small scale so that an overall picture can be seen. The triangle below shows how scale minimizes and maximizes the size of the triangle.

2.5 TRANSFORMATIONS OF 2D GEOMETRIC FIGURES

A two-dimensional figure is a figure that has only two dimensions i.e. length and breadth.

We can change a two-dimensional figure, by adding to it height or depth.

A three-dimensional figure is a figure that has three dimensions i.e. length, breadth and height or depth.

2.5.1 Symmetry

When a shape is symmetrical, it means that if we cut through the centre of the shape, both sides of the shape will be exactly the same size and shape. If we were to draw this triangle on a piece of paper and fold the paper in half, then the shape will be identical on both sides of the folding line.