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
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Fundamentals of Nanoparticles: Classifications, Synthesis