4.3 Transformations

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 ${90}^{\circ }$ or ${180}^{\circ }$ around the origin?

Investigation : rotation of a point by ${90}^{\circ }$

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

What do you notice about the $x$ -coordinates? What do you notice about the $y$ -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 ${180}^{\circ }$

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

What do you notice about the $x$ -coordinates? What do you notice about the $y$ -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 ${}^{\circ }$ clockwise rotation: The image of a point P $(x;y)$ rotated clockwise through 90 ${}^{\circ }$ around the origin is P' $(y;-x)$ . We write the rotation as $(x;y)\to (y;-x)$ .
• 90 ${}^{\circ }$ anticlockwise rotation: The image of a point P $(x;y)$ rotated anticlockwise through 90 ${}^{\circ }$ around the origin is P' $(-y;x)$ . We write the rotation as $(x;y)\to (-y;x)$ .
• 180 ${}^{\circ }$ rotation: The image of a point P $(x;y)$ rotated through 180 ${}^{\circ }$ around the origin is P' $(-x;-y)$ . We write the rotation as $(x;y)\to (-x;-y)$ .

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 ${}^{\text{'}}$ with A(4;2) and A ${}^{\text{'}}$ (-2;4)
   2. OB is rotated to OB ${}^{\text{'}}$ with B(-2;5) and B ${}^{\text{'}}$ (5;2)
   3. OC is rotated to OC ${}^{\text{'}}$ with C(-1;-4) and C ${}^{\text{'}}$ (1;4)
2. Copy $\Delta$ XYZ onto squared paper. The co-ordinates are given on the picture.
   1. Rotate $\Delta$ XYZ anti-clockwise through an angle of 90 ${}^{\circ }$ about the origin to give $\Delta$ X ${}^{\text{'}}$ Y ${}^{\text{'}}$ Z ${}^{\text{'}}$ . Give the co-ordinates of X ${}^{\text{'}}$ , Y ${}^{\text{'}}$ and Z ${}^{\text{'}}$ .
   2. Rotate $\Delta$ XYZ through 180 ${}^{\circ }$ about the origin to give $\Delta$ X ${}^{\text{'}}$ ${}^{\text{'}}$ Y ${}^{\text{'}}$ ${}^{\text{'}}$ Z ${}^{\text{'}}$ ${}^{\text{'}}$ . Give the co-ordinates of X ${}^{\text{'}}$ ${}^{\text{'}}$ , Y ${}^{\text{'}}$ ${}^{\text{'}}$ and Z ${}^{\text{'}}$ ${}^{\text{'}}$ .

   

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 $k$ ?

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.

What do you notice about the $x$ -coordinates? What do you notice about the $y$ -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

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.

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 $(x_1;y_1)$ , T $(x_2;y_2)$ , U $(x_3;y_3)$ . $▵$ S'T'U' is an enlargement through the origin of $▵$ STU by a factor of $c$ ( $c>0$ ).

• $▵$ STU is a reduction of $▵$ S'T'U' by a factor of $c$ .
• $▵$ S'T'U' can alternatively be seen as an reduction through the origin of $▵$ STU by a factor of $\frac{1}{c}$ . (Note that a reduction by $\frac{1}{c}$ is the same as an enlargement by $c$ ).
• The vertices of $▵$ S'T'U' are S' $(c{x}_1;c{y}_1)$ , T' $(c{x}_2,c{y}_2)$ , U' $(c{x}_3,c{y}_3)$ .
• The distances from the origin are OS' = $c$ OS, OT' = $c$ OT and OU' = $c$ OU.

Transformations

1. Copy polygon STUV onto squared paper and then answer the following questions.

   

   1. What are the co-ordinates of polygon STUV?
   2. Enlarge the polygon through the origin by a constant factor of $c=2$ . 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 $k$ 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 $k$ .
   3. If the area of $▵$ ABC is $m$ times the area of $▵$ A'B'C', what is $m$ ?

   

3. 

   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 $c=3$ , 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 ${}^{\circ }$ around the origin.
   5. Find the inclination of OM”.