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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 or 180 around the origin?

Investigation : rotation of a point by 90

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

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 clockwise rotation: The image of a point P ( x ; y ) rotated clockwise through 90 around the origin is P' ( y ; - x ) . We write the rotation as ( x ; y ) ( y ; - x ) .
  • 90 anticlockwise rotation: The image of a point P ( x ; y ) rotated anticlockwise through 90 around the origin is P' ( - y ; x ) . We write the rotation as ( x ; y ) ( - y ; x ) .
  • 180 rotation: The image of a point P ( x ; y ) rotated through 180 around the origin is P' ( - x ; - y ) . We write the rotation as ( x ; y ) ( - 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 ' 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 Δ XYZ onto squared paper. The co-ordinates are given on the picture.
    1. Rotate Δ XYZ anti-clockwise through an angle of 90 about the origin to give Δ X ' Y ' Z ' . Give the co-ordinates of X ' , Y ' and Z ' .
    2. Rotate Δ XYZ through 180 about the origin to give Δ X ' ' Y ' ' Z ' ' . Give the co-ordinates of X ' ' , Y ' ' and Z ' ' .

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 ( 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 1 c . (Note that a reduction by 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 ?
    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 around the origin.
    5. Find the inclination of OM”.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Other chapter Q/A we can ask
Moahammedashifali Reply

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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