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.
Find the inclination of OM”.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?