# 4.3 Transformations

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## 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.

 Point $x$ -coordinate $y$ -coordinate A B C D E F G H

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.

 Point $x$ -coordinate $y$ -coordinate A B C D E F G H

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

## 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.

 Point $x$ -coordinate $y$ -coordinate A B C D E F G H

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.

 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 $\left({x}_{1};{y}_{1}\right)$ , T $\left({x}_{2};{y}_{2}\right)$ , U $\left({x}_{3};{y}_{3}\right)$ . $▵$ 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' $\left(c{x}_{1};c{y}_{1}\right)$ , T' $\left(c{x}_{2},c{y}_{2}\right)$ , U' $\left(c{x}_{3},c{y}_{3}\right)$ .
• 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 $m$ ?
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”.

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