For example, find the equation of the tangent to the circle at point
$(1,1)$ . The centre of the circle is at
$(0,0)$ . The equation of the circle is
${x}^{2}+{y}^{2}=2$ .
Any line
$OP$ is drawn (not necessarily in the first quadrant), making an angle of
$\theta $ degrees with the
$x$ -axis. Using the co-ordinates of
$P$ and the angle
$\alpha $ , calculate the co-ordinates of
${P}^{\text{'}}$ , if the line
$OP$ is rotated about the origin through
$\alpha $ degrees.
$P$
$\alpha $
1.
(2, 6)
60
${}^{\circ}$
2.
(4, 2)
30
${}^{\circ}$
3.
(5, -1)
45
${}^{\circ}$
4.
(-3, 2)
120
${}^{\circ}$
5.
(-4, -1)
225
${}^{\circ}$
6.
(2, 5)
-150
${}^{\circ}$
Characteristics of transformations
Rigid transformations like translations, reflections, rotations and glide reflections preserve shape and size, and that enlargement preserves shape but not size.
Geometric transformations:
Draw a large 15
$\times $ 15 grid and plot
$\u25b5ABC$ with
$A(2;6)$ ,
$B(5;6)$ and
$C(5;1)$ . Fill in the lines
$y=x$ and
$y=-x$ .
Complete the table below , by drawing the images of
$\u25b5ABC$ under the given transformations. The first one has been done for you.
A transformation that leaves lengths and angles unchanged is called a rigid transformation. Which of the above transformations are rigid?
Exercises
$\Delta ABC$ undergoes several transformations forming
$\Delta {A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ . Describe the relationship between the angles and sides of
$\Delta ABC$ and
$\Delta {A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ (e.g., they are twice as large, the same, etc.)
Transformation
Sides
Angles
Area
Reflect
Reduce by a scale factor of 3
Rotate by 90
${}^{\circ}$
Translate 4 units right
Enlarge by a scale factor of 2
$\Delta DEF$ has
$\widehat{E}={30}^{\circ}$ ,
$DE=4\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ ,
$EF=5\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ .
$\Delta DEF$ is enlarged by a scale factor of 6 to form
$\Delta {D}^{\text{'}}{E}^{\text{'}}{F}^{\text{'}}$ .
$\Delta XYZ$ has an area of
$6\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{2}$ . Find the area of
$\Delta {X}^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ if the points have been transformed as follows:
$(x,y)\to (x+2;y+3)$
$(x,y)\to (y;x)$
$(x,y)\to (4x;y)$
$(x,y)\to (3x;y+2)$
$(x,y)\to (-x;-y)$
$(x,y)\to (x;-y+3)$
$(x,y)\to (4x;4y)$
$(x,y)\to (-3x;4y)$
Questions & Answers
where we get a research paper on Nano chemistry....?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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?