SAMPLE LESSON: MATHEMATICS

Class: Form 4

Title of Module: Plane Geometry Title of Chapter: Transformation of plane shapes

Title of Lesson: Reflection Duration of Lesson: 90mins

Module: Plane Geometry Topic: Transformation of plane figures Lesson 3: Reflection Objectives: At the end of this lesson you should be able to:  Describe a reflection;  Identify situations of reflection in real life;  Find image of a point, line or plane surface by a reflection;  Perform or demonstrate reflection of plane figures on the coordinate system;

Rationale: When you look yourself in the mirror what do you see? Reflection is everywhere around us, we see reflection in mirrors, glasses, water, shiny surfaces etc. Didactic material you need for this lesson: Grid, Square paper or graph paper Pencil, ruler, set square or protractor

References: 1. GEOMETRY, Eugene D. Nicholas, Mervine L. Edwards, E Henry Garland, Sylvia, A Hoffman, Albert Mamary, William F Palmer (1991), Holt, Rinehart and Winston, Inc. 2. Mathematics 9, M.J. Tipler, J Douglas (2004), Nelson Thornes Ltd. 3. https://mathbitsnotebook.com/Geometry/Transformations/TRTransformationReflection.html; 4. https://www.pinterest.com/pin/422281185487554/ 5. https://www.mathsisfun.com/geometry/reflection.html 6. https://www.google.com/search?q=Reflection+in+Mathematics&oq=Reflection+in+Mathematics &aqs=chrome..69i57j0l7.14483j0j7&sourceid=chrome&ie=UTF­8

Introduction A reflection of an object is the 'flip' of that object over a line, called the line of reflection. In the following picture, the birds are reflected in the water. Their images are flipped over the line created by the edge of the water. Lines of reflection do not have to be horizontal; they can also be vertical or sloped in any direction

Every point is the same distance from the central line.

The image after reflection has the same size as the original object

The central line is called the Mirror Line

Here are some more examples of reflection in real life.

Reflection involves “flipping” or folding an object over a line called the line of reflection. A reflection can be seen, for example, in water (first 2 images), a mirror (third image), or in a shiny surface. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. The original object is called the pre­image, and the reflection is called the image. • The image is usually labeled using a prime symbol, such as A'B'C'. • An object and its reflection have the same shape and size, but the figures face in opposite directions. The objects appear as if they are mirror reflections, with right and left reversed. Take a look again at the images above showing reflections.

Reflection in the coordinate Plane. Reflection in the coordinate plane depends on the line of reflection

I­ Reflection in the x-axis

Activity 1:

Consider the figure below:

1. Write out the coordinates of the points A, B and C. 2. Write out the coordinates of the points A’, B’, C’. 3. Compare the x­coordinates of A and A’; B and B’; C and C’. 4. Compare the y­coordinates of A and A’; B and B’; C and C’. 5. Compare the distances of A and A’; B and B’, C and C’ from the x­axis

Solution of Activity 1: 1. A(­2, 1); B(1, 4); C(3, 2) 2. A’(­2, ­1); B’(1, ­4); C’(3, ­2) 3. The x coordinates of A and A’ are the same. The x­coordinates of B and B’ are the same. The x­coordinates of C and C’ are the same. 4. The y­coordinate of A is opposite to the y­coordinate of A’; The y­coordinate of the point B is opposite the y­coordinate of B’; The y­coordinate of the point C is opposite they­coordinate of C’. 5. A is 1 unit from the x­axis and A’ is equally 1 unit. B is 4units and B’ is also 4 units from the x­axis. C is 2units and C’ is also 2 units from the x­axis.

Conclusion For any reflection in the x­axis, when any point is reflected across the x­axis, the x­coordinate remains the same, but the y­coordinate is transformed into its opposite (its sign is changed).

The reflection of the point (x, y) across

the x­axis is the point (x,­y).

II­ Reflection over the y­axis Activity 2: Consider the figure below

1. Write down the coordinates of each of the points A, B and C. 2. Write down the coordinates of each of the points A’; B’; C’. 3. Compare the coordinates of A and A’; of B and B’ and of C and C’. 4. Find the distance of A from the y­ axis. Also find the distance of A’ from the y­axis 5. Repeat 4) for B and B’, C and C’ 6. Given that the image of ABC is A’B’C’, what can you say about the coordinates of a point and the coordinates of its image point? 7. What can you say about the distances of corresponding points?

Solution to activity 2: 1. A (3, 1); B(4, 4) and C(1, 3) 2. A’(­3, 1); B’(­4, 4) and C’(­1, 3) 3. For the coordinates of A and A’, the y­coordinates are the same, the x­coordinates are opposite in sign. This is similar for points B and B’; C and C’. 4. The point A is 3 units form the y­axis and the point A’ is equally 3 units from the y­ axis. 5. The point B is 4 units from the y­axis and B’ is 4units from the y­axis. The point C is 1unit from the y­axis and the point C’ is 1unit from the y­axis.

6. The y­coordinates of the object and that of the image points are the same, while the x­coordinates are opposite of each other. 7. The distance of each point and its corresponding image from the y­axis are the same.

Conclusion: When you reflect a point across the y­axis, the y­coordinate remains the same, but the x­ coordinate is transformed into its opposite (its sign is changed).

The reflection of the point (x, y)

across the y­axis is the point (­x, y).

III­ Reflection over the line y = x

The figure by the side is an example of reflection in the line y = x.

If you consider any point and its corresponding image point, you will discover that: When you reflect a point across the line y = x, the x­coordinate and y­coordinate change places. If you reflect over the line y = ­x, the x­coordinate and y­coordinate change places and are negated (the signs are changed).

The reflection of the point (x,y) across

the line y = x is the point (y, x). &

The reflection of the point (x,y) across the line y = ­x is the point (­y, ­x).

IV­ Reflection over any line

The plane figure ABCD is reflected over the line x = ­3 as seen in the figure

Observe that each point of the reflected image is the same distance from the line of reflection as the corresponding point of the original figure. The line of reflection lies directly in the middle between the original figure and its image.

Notice that each point of the original figure and its image are the same distance away from the line of reflection.

V­ Reflection in a Point:

Reflection in a line exists when a figure is built around a single point called the Point of Reflection and also called the Centre of the figure. For every point in the figure, there is another point found directly opposite it on the other side of the center such that the point of reflection becomes the midpoint of the segment joining the point with its image. Under a point reflection, figures do not change size or shape.

While any point in the coordinate plane may be used as a point of reflection, the most commonly used point is the origin. Assume that the origin is the point of reflection unless told otherwise.

VI- Reflection in the Origin (0, 0)

Triangle A'B'C' is the image of triangle ABC after a point reflection in the origin (0, 0). The origin is the midpoint of any line segment connecting an object point with its corresponding image point. When a point is reflected in the origin, both the x­coordinates and the y­coordinate change sign (their signs change).

In a point reflection in the origin, the image of the point (x, y) is the point (­x,­y).

Conclusion In carrying out Reflection geometrically,

1. Plot the points for the vertices of the object figure; 2. Measure the distance of this point from the mirror line. This distance must be that of the line segment that hits the

mirror line at a right angle.

3. Measure the same distance again on the other side and place a dot. 4. Repeat 2 and 3 for all the vertex points

5. Connect the new dots to form the image figure

Points to Remember  Reflection preserves, distance, shape and size but changes orientation.

 The object and the image face each other.

 It is therefore referred to as Opposite Isometry.

 Most reflections are drawn on the coordinate plane. To draw a reflection, just draw

each point of the image on the opposite side of the line of reflection, making sure

that the image point is the same distance away from the line as the corresponding

object point.

 A reflection is an isometric transformation (rigid transformations) because the

image is the same size and shape as the original object.

 The original object and the image are congruent.

Home work: Use the coordinate system below and triangle ABC to answer questions 1 to3 below.

1. Reflect triangle ABC in the x­axis. Call this image A’B’C’. Write out the coordinates of each image point and draw triangle A’B’C’. 2. Reflect triangle ABC in the y­axis. Call the image A’’B’’C’’. Draw the image on the same coordinate system and write out the coordinates of the image points. 3. Reflect triangle ABC in the line y=­x. Call the image A’’’B’’’C’’’. Draw this image and write out the coordinates. 4. Plot the points D(­2, 1) and E(3, ­2) on the coordinate axis. Draw line segment DE. Reflect the line segment DE in x­axis. Reflect the image D’E’ in the y­axis to have D’’E’’. Give the coordinates of the points D’’ and E’’.

5. Write down the coordinates of a reflection across the x­axis for the points indicated P (-10, -10) ⟶ P’ ( , ); Q (-10, 0) ⟶ Q’ ( , ); R (-7, -10) ⟶ R’ ( , )

GOOD LUCK IN YOUR STUDY