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:
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?
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?