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Geometry: Transformations (Grade 10) [NCS]

Module by: Free High School Science Texts Project. E-mail the author

Transformations

In this section you will learn about how the co-ordinates of a point change when the point is moved horizontally and vertically on the Cartesian plane. You will also learn about what happens to the co-ordinates of a point when it is reflected on the xx-axis, yy-axis and the line y=xy=x.

Translation of a Point

When something is moved in a straight line, we say that it is translated. What happens to the co-ordinates of a point that is translated horizontally or vertically?

Discussion : Translation of a Point Vertically

Complete the table, by filling in the co-ordinates of the points shown in the figure.

Figure 1
Figure 1 (MG10C14_022.png)

Table 1
Point xx co-ordinate yy co-ordinate
A    
B    
C    
D    
E    
F    
G    

What do you notice about the xx co-ordinates? What do you notice about the yy co-ordinates? What would happen to the co-ordinates of point A, if it was moved to the position of point G?

When a point is moved vertically up or down on the Cartesian plane, the xx co-ordinate of the point remains the same, but the yy co-ordinate changes by the amount that the point was moved up or down.

For example, in Figure 2 Point A is moved 4 units upwards to the position marked by G. The new xx co-ordinate of point A is the same (xx=1), but the new yy co-ordinate is shifted in the positive yy direction 4 units and becomes yy=-2+4=2. The new co-ordinates of point A are therefore G(1;2). Similarly, for point B that is moved downwards by 5 units, the xx co-ordinate is the same (x=-2,5x=-2,5), but the yy co-ordinate is shifted in the negative yy-direction by 5 units. The new yy co-ordinate is therefore yy=2,5 -5=-2,5.

Figure 2: Point A is moved 4 units upwards to the position marked by G. Point B is moved 5 units downwards to the position marked by H.
Figure 2 (MG10C14_023.png)

Tip:

If a point is shifted upwards, the new yy co-ordinate is given by adding the shift to the old yy co-ordinate. If a point is shifted downwards, the new yy co-ordinate is given by subtracting the shift from the old yy co-ordinate.

Discussion : Translation of a Point Horizontally

Complete the table, by filling in the co-ordinates of the points shown in the figure.

Figure 3
Figure 3 (MG10C14_024.png)

Table 2
Point xx co-ordinate yy co-ordinate
A    
B    
C    
D    
E    
F    
G    

What do you notice about the xx co-ordinates? What do you notice about the yy co-ordinates?

What would happen to the co-ordinates of point A, if it was moved to the position of point G?

When a point is moved horizontally left or right on the Cartesian plane, the yy co-ordinate of the point remains the same, but the xx co-ordinate changes by the amount that the point was moved left or right.

For example, in Figure 4 Point A is moved 4 units right to the position marked by G. The new yy co-ordinate of point A is the same (yy=1), but the new xx co-ordinate is shifted in the positive xx direction 4 units and becomes xx=-2+4=2. The new co-ordinate of point A at G is therefore (2;1). Similarly, for point B that is moved left by 5 units, the yy co-ordinate is the same (y=-2,5y=-2,5), but the xx co-ordinate is shifted in the negative xx-direction by 5 units. The new xx co-ordinate is therefore xx=2,5 -5=-2,5. The new co-ordinates of point B at H is therefore (-2,5;1).

Figure 4: Point A is moved 4 units to the right to the position marked by G. Point B is moved 5 units to the left to the position marked by H.
Figure 4 (MG10C14_025.png)

Tip:

If a point is shifted to the right, the new xx co-ordinate is given by adding the shift to the old xx co-ordinate. If a point is shifted to the left, the new xx co-ordinate is given by subtracting the shift from the old xx co-ordinate.

Reflection of a Point

When you stand in front of a mirror your reflection is located the same distance (dd) behind the mirror as you are standing in front of the mirror.

Figure 5
Figure 5 (MG10C14_026.png)

We can apply the same idea to a point that is reflected on the xx-axis, the yy-axis and the line y=xy=x.

Reflection on the xx-axis

If a point is reflected on the xx-axis, then the reflection must be the same distance below the xx-axis as the point is above the xx-axis and vice-versa, as though it were a mirror image.

Figure 6: Points A and B are reflected on the xx-axis. The original points are shown with and the reflected points are shown with .
Figure 6 (MG10C14_027.png)
Tip:
When a point is reflected about the xx-axis, only the yy co-ordinate of the point changes.
Exercise 1: Reflection on the xx-axis

Find the co-ordinates of the reflection of the point P, if P is reflected on the xx-axis. The co-ordinates of P are (5;10).

Solution
  1. Step 1. Determine what is given and what is required :

    We are given the point P with co-ordinates (5;10) and need to find the co-ordinates of the point if it is reflected on the xx-axis.

  2. Step 2. Determine how to approach the problem :

    The point P is above the xx-axis, therefore its reflection will be the same distance below the xx-axis as the point P is above the xx-axis. Therefore, yy=-10.

    For a reflection on the xx-axis, the xx co-ordinate remains unchanged. Therefore, xx=5.

  3. Step 3. Write the final answer :

    The co-ordinates of the reflected point are (5;-10).

Reflection on the yy-axis

If a point is reflected on the yy-axis, then the reflection must be the same distance to the left of the yy-axis as the point is to the right of the yy-axis and vice-versa.

Figure 7: Points A and B are reflected on the yy-axis. The original points are shown with and the reflected points are shown with .
Figure 7 (MG10C14_028.png)
Tip:
When a point is reflected on the yy-axis, only the xx co-ordinate of the point changes. The yy co-ordinate remains unchanged.
Exercise 2: Reflection on the yy-axis

Find the co-ordinates of the reflection of the point Q, if Q is reflected on the yy-axis. The co-ordinates of Q are (15;5).

Solution
  1. Step 1. Determine what is given and what is required :

    We are given the point Q with co-ordinates (15;5) and need to find the co-ordinates of the point if it is reflected on the yy-axis.

  2. Step 2. Determine how to approach the problem :

    The point Q is to the right of the yy-axis, therefore its reflection will be the same distance to the left of the yy-axis as the point Q is to the right of the yy-axis. Therefore, xx=-15.

    For a reflection on the yy-axis, the yy co-ordinate remains unchanged. Therefore, yy=5.

  3. Step 3. Write the final answer :

    The co-ordinates of the reflected point are (-15;5).

Reflection on the line y=xy=x

The final type of reflection you will learn about is the reflection of a point on the line y=xy=x.

Casestudy : Reflection of a point on the line y=xy=x

Figure 8
Figure 8 (MG10C14_029.png)

Study the information given and complete the following table:

Table 3
  Point Reflection
A (2;1) (1;2)
B (-112112;-2) (-2;-11212)
C (-1;1)  
D (2;-3)  

What can you deduce about the co-ordinates of points that are reflected about the line y=xy=x?

The xx and yy co-ordinates of points that are reflected on the line y=xy=x are swapped around, or interchanged. This means that the xx co-ordinate of the original point becomes the yy co-ordinate of the reflected point and the yy co-ordinate of the original point becomes the xx co-ordinate of the reflected point.

Figure 9: Points A and B are reflected on the line y=xy=x. The original points are shown with and the reflected points are shown with .
Figure 9 (MG10C14_030.png)
Tip:
The xx and yy co-ordinates of points that are reflected on the line y=xy=x are interchanged.
Exercise 3: Reflection on the line y=xy=x

Find the co-ordinates of the reflection of the point R, if R is reflected on the line y=xy=x. The co-ordinates of R are (-5;5).

Solution
  1. Step 1. Determine what is given and what is required :

    We are given the point R with co-ordinates (-5;5) and need to find the co-ordinates of the point if it is reflected on the line y=xy=x.

  2. Step 2. Determine how to approach the problem :

    The xx co-ordinate of the reflected point is the yy co-ordinate of the original point. Therefore, xx=5.

    The yy co-ordinate of the reflected point is the xx co-ordinate of the original point. Therefore, yy=-5.

  3. Step 3. Write the final answer :

    The co-ordinates of the reflected point are (5;-5).

Rules of Translation

A quick way to write a translation is to use a 'rule of translation'. For example (x;y)(x+a;y+b)(x;y)(x+a;y+b) means translate point (x;y) by moving a units horizontally and b units vertically.

So if we translate (1;2) by the rule (x;y)(x+3;y-1)(x;y)(x+3;y-1) it becomes (4;1). We have moved 3 units right and 1 unit down.

Translating a Region

To translate a region, we translate each point in the region.

Example

Region A has been translated to region B by the rule: (x;y)(x+4;y+2)(x;y)(x+4;y+2)

Figure 10
Figure 10 (MG10C14_031.png)

Discussion : Rules of Transformations

Work with a friend and decide which item from column 1 matches each description in column 2.

Table 4
Column 1 Column 2
.(x;y)(x;y-3).(x;y)(x;y-3)        a reflection on x-y line
. ( x ; y ) ( x - 3 ; y ) . ( x ; y ) ( x - 3 ; y ) a reflection on the x axis
. ( x ; y ) ( x ; - y ) . ( x ; y ) ( x ; - y ) a shift of 3 units left
. ( x ; y ) ( - x ; y ) . ( x ; y ) ( - x ; y ) a shift of 3 units down
. ( x ; y ) ( y ; x ) . ( x ; y ) ( y ; x ) a reflection on the y-axis
Transformations
  1. Describe the translations in each of the following using the rule (x;y) (...;...)
    Figure 11
    Figure 11 (mg10c14_4.png)
    1. From A to B
    2. From C to J
    3. From F to H
    4. From I to J
    5. From K to L
    6. From J to E
    7. From G to H
    Click here for the solution
  2. A is the point (4;1). Plot each of the following points under the given transformations. Give the co-ordinates of the points you have plotted.
    1. B is the reflection of A in the x-axis.
    2. C is the reflection of A in the y-axis.
    3. D is the reflection of B in the line x=0.
    4. E is the reflection of C is the line y=0.
    5. F is the reflection of A in the line y= x
    Click here for the solution
  3. In the diagram, B, C and D are images of polygon A. In each case, the transformation that has been applied to obtain the image involves a reflection and a translation of A. Write down the letter of each image and describe the transformation applied to A in order to obtain the image.
    Figure 12
    Figure 12 (mg10c14_5.png)
    Click here for the solution
Investigation : Calculation of Volume, Surface Area and scale factors of objects
  1. Look around the house or school and find a can or a tin of any kind (e.g. beans, soup, cooldrink, paint etc.)
  2. Measure the height of the tin and the diameter of its top or bottom.
  3. Write down the values you measured on the diagram below:
    Figure 13
    Figure 13 (MG10C14_034.png)
  4. Using your measurements, calculate the following (in cm22, rounded off to 2 decimal places):
    1. the area of the side of the tin (i.e. the rectangle)
    2. the area of the top and bottom of the tin (i.e. the circles)
    3. the total surface area of the tin
  5. If the tin metal costs 0,17 cents/cm22, how much does it cost to make the tin?
  6. Find the volume of your tin (in cm33, rounded off to 2 decimal places).
  7. What is the volume of the tin given on its label?
  8. Compare the volume you calculated with the value given on the label. How much air is contained in the tin when it contains the product (i.e. cooldrink, soup etc.)
  9. Why do you think space is left for air in the tin?
  10. If you wanted to double the volume of the tin, but keep the radius the same, by how much would you need to increase the height?
  11. If the height of the tin is kept the same, but now the radius is doubled, by what scale factor will the:
    1. area of the side surface of the tin increase?
    2. area of the bottom/top of the tin increase?

Summary

  • Surface area of rectangular and triangular prisms as well as cylinders
  • Volume of prisms
  • Similarity of polygons Two polygons are similar if: 1. their corresponding angles are equal, and 2. the ratios of corresponding sides are equal. all squares are similar
  • Distance formula
  • Gradient formula
  • Midpoint formula
  • Parallel and perpendicular lines: determining them
  • Translation of a point Vertically changes y co-ordinate Horizontally changes x co-ordinate
  • Reflection of a point x-axis When a point is reflected about the x-axis, only the y co-ordinate of the point changes y-axis When a point is reflected on the y-axis, only the x co-ordinate of the point changes. The y co-ordinate remains unchanged. on line y = x The x and y co-ordinates of points that are reflected on the line y = x are interchanged.
  • To translate a region, translate each point in the region

End of Chapter Exercises

  1. Using the rules given, identify the type of transformation and draw the image of the shapes.
    1. (x;y)(x+3;y-3)
      Figure 14
      Figure 14 (MG10C14_037.png)
    2. (x;y)(x-4;y)
      Figure 15
      Figure 15 (MG10C14_038.png)
    3. (x;y)(y;x)
      Figure 16
      Figure 16 (MG10C14_039.png)
    4. (x;y)(-x;-y)
      Figure 17
      Figure 17 (MG10C14_040.png)
    Click here for the solution
  2. PQRS is a quadrilateral with points P(0; −3) ; Q(−2;5) ; R(3;2) and S(3;–2) in the Cartesian plane.
    1. Find the length of QR.
    2. Find the gradient of PS.
    3. Find the midpoint of PR.
    4. Is PQRS a parallelogram? Give reasons for your answer.
    Click here for the solution
  3. A(–2;3) and B(2;6) are points in the Cartesian plane. C(a;b) is the midpoint of AB. Find the values of a and b.
    Click here for the solution
  4. Consider: Triangle ABC with vertices A (1; 3) B (4; 1) and C (6; 4):
    1. Sketch triangle ABC on the Cartesian plane.
    2. Show that ABC is an isoceles triangle.
    3. Determine the co-ordinates of M, the midpoint of AC.
    4. Determine the gradient of AB.
    5. Show that the following points are collinear: A, B and D(7;-1)
    Click here for the solution
  5. In the diagram, A is the point (-6;1) and B is the point (0;3)
    Figure 18
    Figure 18 (MG10C14_5.png)
    1. Find the equation of line AB
    2. Calculate the length of AB
    3. A’ is the image of A and B’ is the image of B. Both these images are obtain by applying the transformation: (x;y)(x-4;y-1). Give the coordinates of both A’ and B’
    4. Find the equation of A’B’
    5. Calculate the length of A’B’
    6. Can you state with certainty that AA'B'B is a parallelogram? Justify your answer.
    Click here for the solution
  6. The vertices of triangle PQR have co-ordinates as shown in the diagram.
    Figure 19
    Figure 19 (mg10c14_6.png)
    1. Give the co-ordinates of P', Q' and R', the images of P, Q and R when P, Q and R are reflected in the line y=x.
    2. Determine the area of triangle PQR.
    Click here for the solution

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