Transformation Geometry MODULE - 6 Transformation Geometry

Transformation geometry is the geometry of moving points and shapes. The two types of transformations dealt with in this module are: Translations of p units horizontally and q units vertically (LO 3 AS 4) Reflections in the x – axis, the y – axis or the line y = x. (LO 3 AS 4)

TRANSLATIONS A translation is a horizontal or vertical “slide” from one position to another. The object translated does not change its shape or size.

Example 1 Consider figure ABCD in the diagram below. Notice that figures ABCD has been translated 7 units to the right to form an image called A'B'C'D'. Notice that the four corner points have moved 7 units to the right: A has translated 7 units to the right to form its image A‘ B has translated 7 units to the right to form its image B‘ C has translated 7 units to the right to form its image C‘ D has translated 7 units to the right to form its image D‘ Clearly, figure ABCD is congruent to A'B'C'D' since they are identical in size and shape.

Example 2 ABCD has translated 8 units downward to form its image A'B'C'D'. Notice that the four corner points have moved 8 units downward: A has translated 8 units downward to form its image A‘ B has translated 8 units downward to form its to its image B‘ C has translated 8 units downward to form its to its image C‘ D has translated 8 units downward to form its to its image D‘ Clearly, figure ABCD is congruent to A'B'C'D' since theyare identical in size and shape.

Example 3 In this example, ABCD has first translated 8 units to the right and then 6 units downwards. Notice that the four corner points have moved 8 units to the right and then 6 units downwards. A has moved 8 units to the right and then 6 units downward to form A‘ B has moved 8 units to the right and then 6 units downward to form B‘ C has moved 8 units to the right and then 6 units downward to form C‘ D has moved 8 units to the right and then 6 units downward to form D‘ Clearly, figure ABCD is congruent to A'B'C'D' since they are identical in size and shape.

LET’S TRY!!!!!!

Translate figure PQRS as follows: 4 units to the left, then 7 units downward and finally 2 units to the right. Draw the image PQRS on the grid.

TRANSLATION OF POINTS Before dealing with the translation of points, we will briefly revise the plotting of points in the Cartesian Number Plane. We refer to ( ; ) as the coordinates of the point represented in the Cartesian Number Plane.

The first coordinate( - coordinate) represents the distance of the point from the vertical y -axis as measured from the -axis. The second coordinate (y-coordinate) represents the distance of the point from the horizontal -axis as measured from the y-axis.

Example 1 Consider the following points represented in the Cartesian Number Plane.

Consider point A (4; 6). The point is 4 units to the right of the vertical y - axis as measured from the x - axis and 6 units above the horizontal x-axis as measured from the vertical y - axis.

LET US TRY THE FOLLOWING:

1. Describe the position of the following points from the vertical and horizontal axes. Make use of the grid on the previous page. (a) K (7; 3) (b) M (5; - 3) (c) D (- 5; 4) (d) E (0; 2) (e) H (3; 0) (f) A (0; - 5) (g) C (- 4; 0)

2, On the following grid, plot the following points: K (3; 5), L (2;- 7), M(- 4; 8), N(- 5;- 6), P (7; 0), Q (0; - 4), R (- 3; 0), S (0; 6)

Example 2 In each of the following diagrams, a point has been translated by a horizontal move followed by a vertical move to form its image.

Describe the translation and then represent the translation in mathematical notation (algebraically).

(a) Point P moved right by 3 units and then downwards by 4 units to form P', the image of P. The x-coordinate of P' was obtained by adding 3 to the x-coordinate of P. The y-coordinate of P was obtained by subtracting 4 from the y-coordinate of P. In other words, the image P' is the point P'(1 + 3;5 - 4). We say that P(1 ; 5) has been translated by (3 ; - 4).

We say algebraically that P has been mapped onto P' by the rule:

(b). Point C moved 4 units left and then (b) Point C moved 4 units left and then upwards by 6 units to form C', the image of C. The x-coordinate of C was obtained by subtracting 4 from the x - coordinate of C. The y—coordinate of C was obtained by adding 6 to the y-coordinate of C. In other words, the image C is the point C‘ (2 - 4; - 4 + 6). We say that C (2; - 4) has been translated by (- 4; 6). We say algebraically that C has been mapped onto C' by the rule: (x; y)

(c) Point A did not move horizontally at. all (c) Point A did not move horizontally at all. It just moved 6 units upward. The x - coordinate of A' is the same as A because there is no horizontal movement. The y - coordinate of A' was obtained by adding 5 to the y – coordinate of A. In other words, the image A' is the point A' (3; 1 + 5).

We say that A (3; l) has been translated by (0; 5). We say algebraically that A is mapped onto A’ by the rule (x; y)

To summarize: We translate the point (x; y) to the point (x + a ; y + b) by a translation of (a ; b) Where a is a horizontal move and b is a vertical move. If a > 0, the horizontal translation is to the right. If a < 0, the horizontal translation is to the left. If b > 0, the vertical translation is upward. If b < 0, the vertical translation is downward.

LET US TRY!!

1. Determine the coordinates of the image,. M, of the 1. Determine the coordinates of the image, M, of the point M (- 2; 4) if the translation of M to M' is The translation can also be written as (5; - 6).

2. Determine the translation if the 2. Determine the translation if the point D (3; - 4) is translated to the point E‘ (- 3; 4).

TRANSLATION OF REGIONS A region is a shape made up of many points. We can translate regions without changing their size or shape.

Example 1 Translate region ABCD using the transformation rule). Draw the image A'B'C'D' and indicate the coordinates of the vertices of the newly formed figure. The translation here is (- 8; - 6), i.e. 8 units to the left and 6 units downward. Start with any point, for example, A and then apply the transformation. Plot the new point A'. Algebraically, the coordinates of each vertex of figure A'B'C'D' can be calculated as follows:

A, B, C, D.

EXERCISE 1 1. For each of the following points, determine the image formed under the given translation. Use the axes provided on page 13 and 14 to draw each point and its image. You may use the same grid to plot all the points and images. (a) C' if C (4; 3) and (b) D' if D (- 4; 2) and (c) G' if G (l; - 1) and (d) H' if H (3; 4) and (e) I' if I (- 2; - 2) and (f) J' if J (5; -7) and

2. Determine the following translations in each case 2. Determine the following translations in each case. You may use the axes provided to draw the points and images. (a) K (l;- 2) is mapped onto L (5; 3) (b) K (1; - 2) is mapped onto N (- 3;- 5) (c) K (l; - 2) is mapped onto Q (- 4; 3) (d) K (l; - 2) is mapped onto S (3; - 6) (e) K (l; - 2) is mapped onto W (- 3; 2)

3. On the axes provided, draw with coordinates K (- 3; 5), L (- 4; 1) and N (- l; 2). Now draw the images of formed using the following translation rules: (Name the image Triangle 2) (Name the image Triangle 3) (Name the image Triangle 4)

4. Determine the translation rule in each case:

(a) C is mapped onto D. (b) D is mapped onto E. (c) E is mapped onto F. (d) C is mapped onto F. (e) F is mapped onto C. (f) E is mapped onto C. (g) C is mapped onto A.

REFLECTIONS A reflection is a mirror image of a shape about a line of reflection.

If the page is folded on the vertical line, the football player on the left will be exactly the same as the player on the right. We say that the vertical line of reflection is the vertical axis of symmetry.

If the page is folded on the horizontal line, the top part of the symbol will be exactly the same as the bottom part. We say that the horizontal line of reflection is the horizontal axis of symmetry.

REFLECTIONS OF POINTS Reflections about the vertical y-axis

Consider the point A (4; 2) Consider the point A (4; 2). If A is reflected about the y-axis, its image will be the point A'(- 4; 2). Clearly, the line AMA' is perpendicular to the y-axis. Also, it can be seen that AM = MA'

Reflections about the horizontal x-axis Consider the point B (- 2; 3). If B is reflected about the x-axis, its image will be the point B' (- 2; - 3). Clearly, the line BNB' is perpendicular to the x-axis. Also, it can be seen that BN = NB'.

LET US TRY!!!

1. (a). For each point in the diagram. below, draw the reflection of 1. (a) For each point in the diagram below, draw the reflection of the point about the y-axis and indicate the coordinates of the image.

b) Now complete the following table: Point x - coordinate y – coordinate A A’ B B’ C C’ D D’ E E’ (c) What do you notice? (d) Write down an algebraic rule for reflecting the point (x; y) about the y-axis.

2. (a). For each point in the diagram below, draw the 2. (a) For each point in the diagram below, draw the reflection of the point about the x-axis and indicate the coordinates of the image.

(b) Now complete the following table: Point x - coordinate y - coordinate A A’ B B’ C C’ D D’ E E’ (c) What do you notice? (d) Write down an algebraic rule for reflecting the point (x; y) about the x - axis.

Reflections about the line y = x Consider the point B (- 2; 3). If B is reflected about the slanted line y = x, its image will be the point B'( 3; - 2). Clearly, the line BMB' is perpendicular to the line y = x. Also, it can be seen that BM = MB'.

LET US TRY!!

1. (a). For each point in the diagram 1. (a) For each point in the diagram below, draw the reflection of the point about the line y = x and indicate the coordinates of the image.

Point x - coordinate y - coordinate (b) Now complete the following table: Point x - coordinate y - coordinate A A’ B B’ C C’ D D’ E E’ (c) What do you notice? (d) Write down an algebraic rule for reflecting the point (x; y) about the line y = x.

Summary of the rules for reflection Reflection about they-axis: (The first coordinates differ in sign) Reflection about the x-axis: (The second coordinates differ in sign) Reflection about the line y = x: (The first and second coordinates have interchanged)

LET US TRY!!!

Without plotting the point C (3; - 2), determine the coordinates of the image of C if C is reflected about the following lines: (a) y - axis (b) x - axis (c) y = x

REFLECTION OF REGIONS Example

(a). Reflect the region ABCDE about the y-axis to form the image FGHIJ (a) Reflect the region ABCDE about the y-axis to form the image FGHIJ. Determine the rule describing the transformation. (b) Reflect the region FGHIJ about the x-axis to form the image KLMNP. Determine the rule describing the transformation. (c) Reflect the region FGHIJ about the line y = x to form the image QRSTU. Determine the rule describing the transformation.

The first coordinates differ in sign. The coordinates of FGHIJ are: The rule for reflecting ABCDE about the y-axis is The first coordinates differ in sign. The coordinates of FGHIJ are: A (4; 5) F (- 4; 5) B (5; 4) G (- 5; 4) C (5; 1) H (- 5; 1) D (3; l) I (- 3; 1) E (3; 4) J (- 3; 4)

The rule for reflecting FGHIJ about the x-axis is The second coordinates differ in sign. The coordinates of KLMNP are: F (- 4; 5) K (- 4; - 5) G (- 5; 4) L (- 5; - 4) H (- 5; 1) M (- 5; - 1) I (- 3; 1) N (- 3; - I) J (- 3; 4) P (- 3; - 4)

(c) The rule for reflecting FGHIJ about the line y = x is: (x; y) The first and second coordinates are turned around. The coordinates of QRSTU are: F (- 4; 5) H (- 5; 1) J (- 3; 4) On the grid provided below, ABC draw where the coordinates of the vertices are A (- 4; 1), B (-1; 1) and C (- 4; 5). Draw the reflections of ABC about the y-axis, x-axis and the line y = x. Indicate the coordinates of the vertices of the images.

Let us try!!

EXERCISE 2 1. (a) Write down the coordinates of the image of the point P (3; 4) if it is reflected about the x-axis, y-axis and the line y = x. (b)Write down the coordinates of the image of the point P (- 3; - 2) if it is reflected about the x-axis, y-axis and the line y = x. 2. Write down the line of reflection in each case: (a) B (3; 2) is mapped onto B‘ (- 3; 2) (b) B (3; 2) is mapped onto D (3; - 2) (c) B (3; 2) is mapped onto C (2; 3) (d) C (- 4; 5) is mapped onto C' (5; - 4) (e) C (- 4; 5) is mapped onto E (4; 5) (f) C (- 4; 5) is mapped onto F (- 4; - 5)

3. Determine the coordinates of the image formed when K(- 7 ; 2) is transformed using the given rule. State the type of transformation in each case. (a) (x; y) (x; -y) (b) (x; y) (-x; y) (c) (x; y) y; x) (d) (x; y)(x + 3 ; y – 2) 4. On the axes provided on the next page, draw KLM with coordinates K (-2; 4), L (- 3; 1) and M (- l; 3). Now draw the images of KLM formed using the following transformation rules. Indicate the coordinates of the vertices of each image. Do the first two, middle two and last two transformations on the same set of axes (a) (x; y) (x; - y) (call the image Triangle 2) (b) (x; y) (- x; y) (call the image Triangle 3)

(c)(x; y) (x + 2; y)(call the image Triangle 4) (d)(x; y) (x; y – 2)(call the image Triangle 5)

(e)(x; y) (x + 4; y - 1) (call the image Triangle 6) (f)(x; y) (y ; x) (call the image Triangle 7)

COMBINED TRANSFORMATIONS The movement of a shape through two or more transformations is called a combined transformation. A shape may be translated and then reflected, or translated twice or even reflected twice. In a combined transformation, each transformation can be described separately or as a single rule as will be illustrated in the following examples.

Glide Reflection Example 1 A glide reflection is a combined transformation involving a translation and a reflection where the translation moves parallel to the line of reflection. It may also be a reflection followed by a translation which moves parallel to the line of reflection. Figure ABCDE is mapped onto its image A'B'C'D'E.

(a) Describe algebraically the two transformations needed for ABCDE to be mapped onto its image A'B'C'D'E'

(b) Create a single rule to transform ABCDE to its image and then show that this rule actually forms the coordinates of the image A'B'C'D'E.

(a) The first transformation is a translation involving a movement of 8 units to the left with no vertical movement, i.e. a translation of (- 8; 0) or the rule A(x; y) (x - 8; y). A newly formed dotted figure appears. The second transformation is a reflection of the newly formed dotted figure about the x-axis. This means that the signs, of the second coordinates of the dotted figure and figure A'B'C'D'E', will be different. The rule will therefore be (x; y) (x; - y). Using the coordinates of the newly formed doted figure, the rule can be seen as (x —8; y) A' (x - 8; - y).

The single rule which transforms ABCD into ABCD!E is therefore A(x; y) A' (x – 8; - y) We can check whether this single rule works: A (4; 5) A (4 – 8; - 5) = A'(- 4; - 5) which is true. B (3; 4) B (3 - 8; - 4) = B' (- 5; - 4) which is true. C (3; 1) C (-3 – 8; - 1) = C' (- 5; - 1) which is true. D (5; 1) D (5 – 8; - 1) = D' (- 3; - 1) which is true. E (5; 4) E (5 – 8; - 4) = E' (- 3; - 1) which is true. Notice that in this example, the translation moved parallel to the x-axis, which is the line of reflection. The transformation is therefore a glide reflection.

Example 2 In this example, we will deal with a combined transformation involving two reflections, one about the y-axis and the other about the line y = x. Figure ABCDE is mapped onto its image A'B'C'D'E'.

(a) Describe algebraically the two transformations (a) Describe algebraically the two transformations needed for ABCDE to be mapped onto its image A'B'C'D'E'. (b) Create a single rule to transform ABCDE to its image and then show that this rule actually forms the coordinates of the image A'B'C'D'E'.

(a) The first transformation is a reflection about the y-axis. This means that the signs of the x-coordinates of figure ABCDE and the dotted figure will be different. The algebraic rule will therefore be A(x; y) (- x; y). The second transformation will be a reflection of the dotted figure about the line y = x to form A'B'C'D'E'. This means that the first and second coordinates will interchange. The rule will therefore be A (- x; y) (y ;- x). (b) The single rule which transforms ABCD into A'B'C'D'E' is therefore A(x; y) A' (y; - x). We can check whether this single rule works: A (4; 5) A' (5; - 4) which is true. B (3; 4) B' (4; - 3) which is true. C (3; 1) C'(1; -3) which is true. D (5; 1) D'(l; - 5) which is true. E (5; 4) E'(4; - 5) which is true.

EXERCISE 3 1.

Refer to the diagram to answer the following questions: (a) Write down an algebraic rule if A is mapped onto B. (b) Write down an algebraic rule if A is mapped onto D. (c) Write down an algebraic rule if D is mapped onto B. (d) Write down an algebraic rule if B is mapped onto C. (e) Write down an algebraic rule if C is mapped onto B. (f) Describe algebraically the two transformations needed for A to be mapped onto image C. (g) Why is this transformation a glide reflection? (h) Create a single rule to transform A to its image C and then show that this rule actually forms the coordinates of the image. (i) Create a single rule to transform D to its image C. (j) On squared paper, draw the reflection of A about the y-axis.

(k) On squared paper, draw the reflection of K about the line y = x.

(1) On squared paper, draw the reflection of D about the line y = x.

2. Refer to the diagram to answer the following questions. (a) Write down an algebraic rule if A is mapped onto B. (b) Write down an algebraic rule if B is mapped onto C. (c) Describe algebraically the two transformations needed for A to be mapped onto image C. (d) Create a single rule to transform A to its image C and then show that this rule actually forms the coordinates of the image.

ASSESSMENT TASKS 1. MIND MAP Create a mind map of the different types of transformations you have studied. You need to include the algebraic notations in your mind map.

Your mind map will be marked according to the following rubric: Assessment criteria No attempt (0) Some attempt(1) Achieved(2) Explained translations Explained reflections Combined transformations Logical presentation Correct transformation rules

2. SHORT PROJECT The local band “Symmetry” will be releasing their first DVD in two month’s time. You have been asked to create the design for the front cover of the DVD. In your work, you must make use of translations and reflections in an original and exciting way. The use of color is essential. You may use any materials to create the design. (colored pens or pencils, pictures from magazines, books or the Internet).

Marks will be awarded according to the following rubric: Assessment criteria No attempt (0) Some attempt(1) Achieved (2)

ASSIGNMENT ON TRANSFORMATIONS QUESTION 1

Use the above grid to answer the following questions: 1. (a) Draw the image of figure A under the transformation (x; y) (-x; y) on the diagram above. Label it B. (b) Name the transformation:_________________________ 2. (a) Draw the image of figure B under the transformation (x; y) (x + 2; y - 2) on the diagram above. Label it C. (b) Name the transformation: _____________________________ 3. (a) Draw the image of figure C under the transformation (x; y) (x - 8; y), on the diagram above. Label it D. (b) Name the transformation:_____________________________ 4. (a) Draw the image of figure D under the transformation (x; y) (x; y - 6) on the diagram above. Label it E. (b) Name the transformation:_____________________________ 5. (a) Draw the image of figure E under the transformation (x; y) (x; - y) on the diagram above. Label it F. (b) Name the transformation:

6. (a) Draw the image of figure A under the transformation (x; y) (y; x) on the diagram above. Label it H. (b) Name the transformation:_________________________ 7. (a) Draw the image of figure F under the transformation (x; y) (- x; y) on the diagram above, Label it G. (b) Name the transformation:_______________________ 8. (a) Describe the two transformations in the combined transformation from B to H. (b) Express this combined transformation as a single rule from B to it’s image H. 9. (a) Describe the two transformations in the combined transformation from C to E. (b) Express this combined transformation as a single rule from C to it’s image E. 10. Consider the combined transformation (x; y) (- x; - y). (a) Name the two types of transformations in this rule. (b) What figure maps onto its image using this combined transformation rule? Explain your answer.

QUESTION 2

1. Write down an algebraic rule to transform figure A to figure C. Use the above diagram in this Question. 1. Write down an algebraic rule to transform figure A to figure C. 2. Describe the two transformations that will transform figure A to figure B. Hence write down a single algebraic rule to describe the transformation of A to B. 3. Describe the two transformations that will transform figure B to figure D. Hence write down a single algebraic rule to describe the transformation of B to D.

QUESTION 3 In the space provided below, by drawing your own set of axes, draw with vertices A (2; 3), B (4; 2) and C (l; 1). Now answer the following questions. 1. Draw A'B'C' the image of under the rule (x; y) (- x; y). Describe the transformation:_____________________________________ 2. Draw A''B''C'', the image of A'B'C' under the rule (x; y) (x; — y). Describe the transformation:__________________________ 3, Draw A'''B'''C''', the image of A'B'C', under the rule (x; y) (y + 3; x). Describe the two transformations used to transform A'B'C' to A'''B'''C'''