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Home , Point (geometry)

Module: Chapter: Lesson: Class: Form 5

The African Institute for Mathematical Sciences 1 OBJECTIVES

• Recognize symmetry on and in nature; • Identify symmetry and symmetrical figures or objects; • Draw symmetrical figures.

The African Institute for Mathematical Sciences 2 Motivation This lesson introduces you to and Point symmetry. We see Symmetry everyday around us both in nature and in man made constructions. Many shapes in nature such as flowers, animals, wings of a butterfly, leaves of trees, human face, reflection of trees in clear water etc are symmetrical. Nature uses symmetry to make things look nice. Designers use symmetry to make things attractive and practical. Buliders also use symmetry in construction, roofing, tilling of the house etc. Symmetry makes things beautiful.

The African Institute for Mathematical Sciences 3 Some examples of symmetry around us

You must have seen the following before. Do they look nice ?

The African Institute for Mathematical Sciences 4 More examples of symmetry around us

The African Institute for Mathematical Sciences 5 Take some time, look around you and identify things that are showing any form of symmetry

The African Institute for Mathematical Sciences 6 Plan of Lesson

A- LINE SYMMETRY

B- POINT SYMMETRY

The African Institute for Mathematical Sciences 7 Material needed for the lesson

For this lesson you need the following: A ruler A pencil A Pen Graph or papers A4 papers or your exercise book Tracing paper

The African Institute for Mathematical Sciences 8 A- LINE SYMMETRY

Objectives: At the end of this lesson, you should be able to: • Recognize line symmetry on shapes and in nature; • Identify line of symmetry and symmetrical figures; • Draw lines of symmetry and symmetrical figures.

The African Institute for Mathematical Sciences 9 Assessment of Pre-requisite Knowledge

Indicate the correct answer for this question. Only one answer is correct. 1. Two plane figures are said to be congruent if they have: A) Same size; B) Same ; C) Same size and same shape.

10 Answer and explanation

A) Is not a sufficient condition because the two shapes can have the same size but do not fit exactly on each other. B) Is equally not a sufficient condition for two shapes to be congruent. They might be of the same shape while one is bigger than the other. C) Same size and same shape. This is the sufficient condition for two shapes to be congruent. If they have the same size and same shape, they will fit on each other without overlap. C) therefore is the correct answer.

The African Institute for Mathematical Sciences 11 More assessment of pre-requisite

2. The conditions for two to be congruent are: (Indicate all correct answers) A) Two sides and the included of one are equal in measure to the corresponding two sides and included angle of the other triangle (SAS); B) All 3 sides of one triangle are equal in to the corresponding 3 sides of the other triangle (SSS); C) Two and the included side of one triangle are equal in size to the corresponding 2 angles and included side (ASA).

The African Institute for Mathematical Sciences 12 Answers:

All the 3 conditions are 3 methods to establish congruency of triangles usually abbreviated as: SSS; SAS; ASA.

The African Institute for Mathematical Sciences 13 Assessment of pre-requisite

3.Two line segments are congruent if (only one answer is correct): A) They are in the same direction; B) They have the same thickness C) They are of the same length.

Answer: Two lines are congruent if they are of the same length.

The African Institute for Mathematical Sciences 14 Assessment of pre-requisite

4. The conditions for 2 right –angled triangles to be congruent are (Only one answer is correct): A) is equal in length B) A of shorter sides are equal in length C) Hypotenuses are equal in length and a pair of shorter sides are equal in length. Answer is C)

The African Institute for Mathematical Sciences 15 Shapes for learning Activity

The African Institute for Mathematical Sciences 16 Learning Activity 1

1. Draw each of the plane shapes above on a piece of paper. 2. Draw a dotted line on each as indicated on the figure. 3. Fold the figure along this line. 4. What can you say about the two halves? 5. Measure the length of corresponding sides. What can you say about their ? 6. Measure corresponding angles, what can you say about their measure?

The African Institute for Mathematical Sciences 17 Solution to activity

Both halves will fit on each other The parts are not mirror images of each without overlap. One is the mirror other. images of the other. The line is a line of The line is not a line of symmetry symmetry

Both halves will fit on each other without In this , the two halves PQBA overlap. They are mirror images of each and ABRS are identical but are not mirror other. The line is a line of symmetry images of each other. the mirror image of point P is T. Therefore The line is not a line of symmetry. The African Institute for Mathematical Sciences 18 Example 1

The figure by the side has one line of symmetry which is the broken line. If the figure is cut along the broken line, the resulting two parts are mirror images of each other. The broken line is the line (axis) of symmetry. This shape has one Line of symmetry. The shape therefore is said to be Symmetrical

The African Institute for Mathematical Sciences 19 Example 2

 If this is cut or folded along any of the dotted lines, the resulting two parts are mirror images of each other.  The Circle has an infinite number of Lines of Symmetry.

The African Institute for Mathematical Sciences 20 More Examples

The African Institute for Mathematical Sciences 21 Summary Line Symmetry is also called Reflection symmetry A line (axis) of Symmetry is a line that divides the shape into two halves such that one half is the mirror image of the other half . This line is also referred to as the Mirror Line. The mirror image of any point is the same distance from the line of symmetry. Corresponding lengths are equal and corresponding angles are equal. A point on the Line of symmetry is its own mirror image Shapes that dare not symmetrical are said to be Asymmetrical

The African Institute for Mathematical Sciences 22 The African Institute for Mathematical Sciences 23 Exercise 1

Consider these figures and answer the questions below: a. b. c.

1. For which figures could a horizontal line of symmetry be drawn? 2. For which figures could a vertical line of symmetry de drawn? 3. State the number of lines of symmetry that can be drawn for figures b) and figure c)? 4. For which figures could a line of symmetry,neither horizontal nor vertical, be drawn?

The African Institute for Mathematical Sciences 24 Exercise 2.

 say whether the dotted line for each of the figures in the next slide is a line of symmetry or not.

The African Institute for Mathematical Sciences 25 The African Institute for Mathematical Sciences 26 Exercise 3

 1. Produce the figure below and use dotted lines to draw other Lines of symmetry.  2. State the number of lines of symmetry for this figure.

The African Institute for Mathematical Sciences 27 Exercise 4

1. Determine if a parallelogram has lines of symmetry 2. If yes how many

The African Institute for Mathematical Sciences 28 Exercise 5

The figure by the side is incomplete, find the images of the points E, F, G and H given that the line p is a line of symmetry Hence complete the figure

The African Institute for Mathematical Sciences 29 Solution

 H is it own image because it is on the line of symmetry.  E and E' are equidistant from the mirror line and the line (EE') is to the line of symmetry.  F' is the image of F by p. F and F' are equidistant from p and the line (FF') is perpendicular to p.  Similarly, G' is the image of G. G and G' are equidistant from p and GG' is perpendicular to P.

The African Institute for Mathematical Sciences 30  LET US NOW LOOK AT POINT SYMMETRY

The African Institute for Mathematical Sciences 31 B-POINT SYMMETRY

Objectives of session:

1. Recognize point symmetry; 2. Demonstrate point symmetry; 3. Observe point symmetry in nature.

The African Institute for Mathematical Sciences 32 Examples of objects with Point Symmetry

The African Institute for Mathematical Sciences 33 More examples of objects with point symmetry

The African Institute for Mathematical Sciences 34 For each of of the shapes above what will happen if you hold in the middle and turn: 1. Through 900. 2. Through 180 3. Through 2700 4. Through 3600

The African Institute for Mathematical Sciences 35 Point Symmetry on Plane shapes

Consider this shape by the side: The shape can be turned holding the centre P fixed. If the point A is at any of the four positions A, B, C or D, the shape will look exactly the same. The point P is the centre. A plane shape has Point symmetry if it fits onto itself two or more times in one turn. The shape beside will fit onto itself 4 times.

The African Institute for Mathematical Sciences 36 The African Institute for Mathematical Sciences 37 Exercise 1: Write down the order of the rotational symmetry if any for each of the shapes below:

The African Institute for Mathematical Sciences 38 Answers: a) has rotational symmetry of order 2 b) has rotational symmetry of order 3 c) has no rotational symmetry d) has rotational symmetry of order 4 e) has rotational symmetry of order 2 f) has rotational symmetry of order 6

The African Institute for Mathematical Sciences 39 Homework: Exercise 1

 The ABCD and IJKH are symmetrical about a line l that is not drawn on the diagram. Figure on next slide 1. Give two points that are on line l. 2. Copy and complete the table with points and their corresponding images with l as a line of symmetry.

A B C D E F G H I J K

The African Institute for Mathematical Sciences 40 The African Institute for Mathematical Sciences 41 Exercise 3:

Homework Exercise 2: State the number of lines of symmetry which could be drawn through the following shapes

The African Institute for Mathematical Sciences 42 Homework Exercise 3

 The table beside gives points and corresponding images A B C I by a point symmetry E F G H O.

Reproduce the figure and construct the points E, F, G and H.

The African Institute for Mathematical Sciences 43 Properties of line and Point symmetry

 Line Symmetry  Point Symmetry 1. A point has as image a 1. A point has as image a point equidistant from the point equidistant from the mirror line; point of symmetry; 2. A line has as image another 2. A line has as image another line line 3. A has as 3. A line segment has as image a line segment of the image a line segment of the same length same length 4. A circle has as image a 4. A circle has as image a circle of the same circle of the same radius 5. An angle has as image an 5. An angle has as image an angle of the same size angle of the same size Size and shape are conserved Size and shape are conserved

The African Institute for Mathematical Sciences 44 References:

1. 9, M J. Tipler & J Douglas (2004), Nelson thornes Ltd;

2. Geometry, Ray C Jurgensen & Alfred J. Donnely & John E. Maier $ Gerald R Rising, (1978), Houghton Miffin company

3. https://www.math-only-math.com/lines-of-symmetry.html 4. https://www.quora.com/What-are-some-amazing-examples-of-symmetry- in-nature

The African Institute for Mathematical Sciences 45 THANKS FOR FOLLOWING UP THIS LESSON THE NEXT LESSON WILL BE ON CONSTRUCTION OF IMAGES UNDER BOTH