# 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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Joseph
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no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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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
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what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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