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Geometry: Transformations (Grade 11)

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

Transformations

Rotation of a Point

When something is moved around a fixed point, we say that it is rotated about the point. What happens to the coordinates of a point that is rotated by 90∘90∘ or 180∘180∘ around the origin?

Investigation : Rotation of a Point by 90∘90∘

Complete the table, by filling in the coordinates of the points shown in the figure.

Table 1
Point xx-coordinate yy-coordinate
A    
B    
C    
D    
E    
F    
G    
H    

Figure 1
Figure 1 (MG11C16_027.png)

What do you notice about the xx-coordinates? What do you notice about the yy-coordinates? What would happen to the coordinates of point A, if it was rotated to the position of point C? What about point B rotated to the position of D?

Investigation : Rotation of a Point by 180∘180∘

Complete the table, by filling in the coordinates of the points shown in the figure.

Table 2
Point xx-coordinate yy-coordinate
A    
B    
C    
D    
E    
F    
G    
H    

Figure 2
Figure 2 (MG11C16_028.png)

What do you notice about the xx-coordinates? What do you notice about the yy-coordinates? What would happen to the coordinates of point A, if it was rotated to the position of point E? What about point F rotated to the position of B?

From these activities you should have come to the following conclusions:

  • 90∘∘ clockwise rotation: The image of a point P(x;y)(x;y) rotated clockwise through 90∘∘ around the origin is P'(y;-x)(y;-x). We write the rotation as (x;y)→(y;-x)(x;y)→(y;-x).
  • 90∘∘ anticlockwise rotation: The image of a point P(x;y)(x;y) rotated anticlockwise through 90∘∘ around the origin is P'(-y;x)(-y;x). We write the rotation as (x;y)→(-y;x)(x;y)→(-y;x).
  • 180∘∘ rotation: The image of a point P(x;y)(x;y) rotated through 180∘∘ around the origin is P'(-x;-y)(-x;-y). We write the rotation as (x;y)→(-x;-y)(x;y)→(-x;-y).
Figure 3
Figure 3 (MG11C16_029.png)
Figure 4
Figure 4 (MG11C16_030.png)
Figure 5
Figure 5 (MG11C16_031.png)

Rotation

  1. For each of the following rotations about the origin: (i) Write down the rule. (ii) Draw a diagram showing the direction of rotation.
    1. OA is rotated to OA'' with A(4;2) and A''(-2;4)
    2. OB is rotated to OB'' with B(-2;5) and B''(5;2)
    3. OC is rotated to OC'' with C(-1;-4) and C''(1;4)
  2. Copy ΔΔXYZ onto squared paper. The co-ordinates are given on the picture.
    1. Rotate ΔΔXYZ anti-clockwise through an angle of 90∘∘ about the origin to give ΔΔX''Y''Z''. Give the co-ordinates of X'', Y'' and Z''.
    2. Rotate ΔΔXYZ through 180∘∘ about the origin to give ΔΔX''''Y''''Z''''. Give the co-ordinates of X'''', Y'''' and Z''''.
    Figure 6
    Figure 6 (MG11C16_032.png)

Enlargement of a Polygon 1

When something is made larger, we say that it is enlarged. What happens to the coordinates of a polygon that is enlarged by a factor kk?

Investigation : Enlargement of a Polygon

Complete the table, by filling in the coordinates of the points shown in the figure. Assume each small square on the plot is 1 unit.

Table 3
Point xx-coordinate yy-coordinate
A    
B    
C    
D    
E    
F    
G    
H    

Figure 7
Figure 7 (MG11C16_033.png)

What do you notice about the xx-coordinates? What do you notice about the yy-coordinates? What would happen to the coordinates of point A, if the square ABCD was enlarged by a factor 2?

Investigation : Enlargement of a Polygon 2

Figure 8
Figure 8 (MG11C16_034.png)

In the figure quadrilateral HIJK has been enlarged by a factor of 2 through the origin to become H'I'J'K'. Complete the following table using the information in the figure.

Table 4
Co-ordinate Co-ordinate' Length Length'
H = (;) H' = (;) OH = OH' =
I = (;) I' = (;) OI = OI' =
J = (;) J' = (;) OJ = OJ' =
K = (;) K' + (;) OK = OK' =

What conclusions can you draw about

  1. the co-ordinates
  2. the lengths when we enlarge by a factor of 2?

We conclude as follows:

Let the vertices of a triangle have co-ordinates S(x1;y1)(x1;y1), T(x2;y2)(x2;y2), U(x3;y3)(x3;y3). ▵▵S'T'U' is an enlargement through the origin of ▵▵STU by a factor of cc (c>0c>0).

  • ▵▵STU is a reduction of ▵▵S'T'U' by a factor of cc.
  • ▵▵S'T'U' can alternatively be seen as an reduction through the origin of ▵▵STU by a factor of 1c1c. (Note that a reduction by 1c1c is the same as an enlargement by cc).
  • The vertices of ▵▵S'T'U' are S'(cx1;cy1)(cx1;cy1), T'(cx2,cy2)(cx2,cy2), U'(cx3,cy3)(cx3,cy3).
  • The distances from the origin are OS' = ccOS, OT' = ccOT and OU' = ccOU.

Figure 9
Figure 9 (MG11C16_035.png)

Transformations

  1. Copy polygon STUV onto squared paper and then answer the following questions.
    Figure 10
    Figure 10 (MG11C16_036.png)
    1. What are the co-ordinates of polygon STUV?
    2. Enlarge the polygon through the origin by a constant factor of c=2c=2. Draw this on the same grid. Label it S'T'U'V'.
    3. What are the co-ordinates of the vertices of S'T'U'V'?
  2. ▵▵ABC is an enlargement of ▵▵A'B'C' by a constant factor of kk through the origin.
    1. What are the co-ordinates of the vertices of ▵▵ABC and ▵▵A'B'C'?
    2. Giving reasons, calculate the value of kk.
    3. If the area of ▵▵ABC is mm times the area of ▵▵A'B'C', what is mm?
    Figure 11
    Figure 11 (MG11C16_037.png)
  3. Figure 12
    Figure 12 (MG11C16_038.png)
    1. What are the co-ordinates of the vertices of polygon MNPQ?
    2. Enlarge the polygon through the origin by using a constant factor of c=3c=3, obtaining polygon M'N'P'Q'. Draw this on the same set of axes.
    3. What are the co-ordinates of the new vertices?
    4. Now draw M”N”P”Q” which is an anticlockwise rotation of MNPQ by 90∘∘ around the origin.
    5. Find the inclination of OM”.

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