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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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what is hormones?
Wellington
Other chapter Q/A we can ask
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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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