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
$(x;y)$ 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
For each of the following rotations about the origin:
(i) Write down the rule.(ii) Draw a diagram showing the direction of rotation.
OA is rotated to OA
${}^{\text{'}}$ with A(4;2) and A
${}^{\text{'}}$ (-2;4)
OB is rotated to OB
${}^{\text{'}}$ with B(-2;5) and B
${}^{\text{'}}$ (5;2)
OC is rotated to OC
${}^{\text{'}}$ with C(-1;-4) and C
${}^{\text{'}}$ (1;4)
Copy
$\Delta $ XYZ onto squared paper. The co-ordinates are given on the picture.
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
${}^{\text{'}}$ .
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
the co-ordinates
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
$({x}_{3};{y}_{3})$ .
$\u25b5$ S'T'U' is an enlargement through the origin of
$\u25b5$ STU by a factor of
$c$ (
$c>0$ ).
$\u25b5$ STU is a reduction of
$\u25b5$ S'T'U' by a factor of
$c$ .
$\u25b5$ S'T'U' can alternatively be seen as an reduction through the origin of
$\u25b5$ 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
$\u25b5$ 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
Copy polygon STUV onto squared paper and then answer the following questions.
What are the co-ordinates of polygon STUV?
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'.
What are the co-ordinates of the vertices of S'T'U'V'?
$\u25b5$ ABC is an enlargement of
$\u25b5$ A'B'C' by a constant factor of
$k$ through the origin.
What are the co-ordinates of the vertices of
$\u25b5$ ABC and
$\u25b5$ A'B'C'?
Giving reasons, calculate the value of
$k$ .
If the area of
$\u25b5$ ABC is
$m$ times the area of
$\u25b5$ A'B'C', what is
$m$ ?
What are the co-ordinates of the vertices of polygon MNPQ?
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.
What are the co-ordinates of the new vertices?
Now draw M”N”P”Q” which is an anticlockwise rotation of MNPQ by 90
${}^{\circ}$ around the origin.
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