Chapter 9 Notes

Notes #36: Translations and Symmetry (Sections 9.1, 9.4)

Transformation:Atransformationof a geometric figure is a change in its position, shape or size.

Preimage:The original figure.

Image:The final figure after a transformation has occurred.

Isometry:When thepreimage and image are congruent. It is a motion that preserves the size and shape of the image as it is transformed.

Translations:

A translation is a transformation of points on a graph in which a set of points slides, or shifts, location. All points affected by a single translation, or slide, must shift the same distance in the same direction.

1.) We are going to translate 3 units to the right and 4 units down by moving points A, B, and C in this manner to points A’, B’, and C,’ respectively.
Write the coordinates of each point:
A A’
B B’
C C’
We describe this translation as (this is the rule for the translation):
(x, y)(______, ______) /
** is called the ______of this translation and
is called the ______of this translation. **
2.) Find the image of the figure under the given translation.
(x, y)  (x – 6, y + 3) /
Complete:
3.)(x, y) (x – 2, y + 1)
a) This translationglides points ____ units left and ____ units up
b) The image of (4, 6) is (____, ____)
c) The preimage of (4, 6) is (____, ____) / 4.) (x, y) (x + 5, y - 2)
a) This translationglides points ______and
______
b) The image of (4, 6) is (____, ____)
c) The preimage of (4, 6) is (____, ____)
State whether the transformation appears to be an isometry.
5.)
/ 6.)

Symmetry:

There are three types of symmetry: reflectional/line, rotational, and point.

Reflectional Symmetry/Line Symmetry
/ How many lines can you draw through the hexagon that make mirror-image congruent halves?
This is an example of reflectional or line symmetry.
One half of the figure is a mirror image of its other half.
Rotational Symmetry
a) b)

/ A figure has rotational symmetry if its own image is created by rotating the image 1800 or less.
Which of these triangles have rotational symmetry?
How many degrees do you need to rotate it to get a symmetric figure?
Point Symmetry
a) b)

c) d) / A figure has point symmetry if it can be rotated about a point 1800 and creates the same image. This is a special type of rotational symmetry.
Which figures have point symmetry?
Tell what type(s) of symmetry each figure has. If it has line symmetry, sketch the line(s) of symmetry. If it has rotational symmetry, state the angle of rotation.
7.) / 8.)
9.) / 10.)



Notes 37: Reflections, Rotations

Reflections:

A reflection is a transformation of points on a graph in which a line acts like a mirror – sending points’ images to new locations on the graph.

Pretend that line m is a mirror. Reflect points A, B, and C about line m. Name their images A’, B’, and C’, respectively.
This reflection in line m is called: _____
We describe individual mappings as:
Reflection of A across m maps A to A’, or: AA’
Write the two other reflections shown:
______
______/
Complete the following. Note the reflections in lines h and k.

7.) Reflection across j stands for ______/ 8)Reflection across h____ / 9) Reflection across j___
10.) Refection across h:____ / 11)Reflection across j: _____ / 12) Reflection across h:

Reflect each image in line p:

13.)

/ 14.)



/ 15.)
/

Write the coordinates of the image of each point by reflection in (a) the x-axis, (b) the y-axis, and (c) the line y = x.

16.) A
a) the x-axis b) the y-axis c) y = x
17.) B
a) the x-axis b) the y-axis c) y = x /
Given points T(2, 4), A(-3, -4), B(0, -4), draw and its reflection image across each line.
18.) x = -3
/ 19.) y = 4

20.) y = x + 2

Rotations:

A rotation is another example of a transformation in which points’ images are sent to new locations on your graph. You will need to imagine sticking your original point to a steering wheel and then rotating the wheel a given number of degrees. Where the point stops is its image under a rotation.

Rotations are described based on the point of rotation (the center of your steering wheel) and the angle of rotation(how many degrees you turn your wheel).

Counterclockwise rotations are indicated by positive degree measures.

Clockwise rotations are indicated by negative degree measures.

Check out these rotations. (I mapped P to P’ and called the center of the wheel H)

900 rotation about H / -900 rotation about H / 5000 rotation about H

Key degree measurements to remember:

______: a quarter of the way around a circle

______: half way around a circle

______: all the way around a circle

Name each image point:

examples:900 Rotation of L about O. Start at point L, rotate the wheel 90 degrees counterclockwise (90 is positive) around point O, and name the point where you would end up: ______

5.) 900 rotation of N about O / 6.) -600 Rotation of M about O. / 7.) 5400 Rotation of P about O. /

State another name for each rotation: (hint: switch direction of rotation and find a new way to get to that spot on the wheel)

8.) 1500 Rotation of P about O9.) -900 Rotation about O 10.) -135 Rotation about O

Notes # 38: Dilations and Tessellations (Sections 9.5, 9.7)

Dilations:

A dilation is a transformation in which make the pre-image proportionally larger (an enlargement) or smaller (a reduction) by a given scale factor. The image will always be similar to the pre-image.

We are going to dilate and shrink it to half its size. This would be a reduction. One such dilation is:
Dilation with center O and scale factor of ½ :
This means: “Graph points A’, B’, and C’ so that
OA’ = ½ OA, OB’ = ½ OB, OC’ = ½ OC”
Steps:
- measure OA and mark its midpoint as A’
- measure OB and mark its midpoint as B’
- measure OC and mark its midpoint as C’
- connect A’, B’, and C’ as a triangle
- is similar to and half the size of
/
11.) Sketch the image under the given dilation: Dilation with a center of O and a scale factor of 2.
You need to construct a triangle that is similar to with a scale factor of 2.

Meaning, your new points X’, Y’, and Z’ must be twice the distance than the given points from O.
This is an example of an enlargement. /
12.) Find the coordinates of the images of A, B, and C by the given dilation.
Dilation with center O and scale factor of 2
A (-2, 4)A’ (____, ____)
B (0, -3)B’ (____, ____)
C (4, 1)C’ (____, ____) /
The blue figure is a dilation image of the green figure. Describe the dilation.
13.)
/ 14.)

15.)

Tessellations

A tessellation is a repeated pattern of figures that completely covers a plane without gaps or overlays. The sum of the measures around any vertex must be 3600.

All triangles tessellate
for triangles n = 3
and 60 is a factor of 360
/ All quadrilaterals tessellate
n = 4
and 90 is a factor of 360.

Determine whether each figure will tessellate a plane.
4.) rhombus / 5.) acute triangle
6.) regular hexagon / 7.) regular decagon
8.) rectangle

Geometry Chapter 9 Study Guide

Reflections:

1. Complete each statement:

a) Reflect across h:
b) Reflect across j
c) Reflect across j / 2. Reflectthe image in
line p

/ 3. Write the coordinates of point A by reflection in the (a) x-axis, (b) y-axis, and
(c) the line y = x

(a) (b) (c)

Translations:

4. Complete:
(x, y) (x - 4, y + 3)
a) Translationglides points ______
______
b) The image of (3, -5) is (____, ____)
c) The preimage of (3, -5) is (____, ____) / 5. Point P and its image P’ are shown. Complete the translation statement.
(x, y) (______, ______)



Rotations:

6. State another name for the rotation:
(a)1200 rotation about O
(b) -2700 rotation about O / 7.Name each image point:
(a) 2100 Rotation of Q about O
(b) 1200 Rotation of M about O /

Dilations: Complete each dilation

8. Center at O; Scale factor ½ :
/



/ 9. Center at O; Scale factor of -2:





Indicate how many lines of symmetry and what type of symmetry for each figure:

H / / / /
10.) / 11.) / 12.) / 13.) / 14.)
Which of these figures will tessellate?
17)
/ 18)

HW#40:Final Exam ReviewName: ______

1.) The dimensions of a
rectangle are 4m by 8m. Find
the length of its diagonal. / 2.) The perimeter of a square
is 20m. Find its area. / 3.) The hypotenuse of a 45-45-90 triangle is 18cm. Find the length of its legs.
4.) The shortest leg of a
30-60-90 triangle is in.
Find the length of the other
two sides of the triangle. / 5.) The longer leg of a
30-60-90 triangle is 21m.
Find the length of the other
two sides of the triangle. / 6.) The perimeter of a regular hexagon is 60cm. Find its area.
7.) Write a trig equation to solve for x.
/ 8.) Write a trig equation to solve for x.
/ 9.) Write a trig equation to solve for x.

10.) The circumference of a circle is in. Find the
area of the circle. / 11.) The perimeter of an equilateral triangle is 27m.
Find its area. / 12.) The diagonals of a rhombus are 10m and 14m. Find its area.
13.) Find the lateral area and volume of a cylinder with
height 3in and radius 4in.
(LA = , V = ) / 14.) Find the surface area of
a sphere with diameter 12m.
(SA = ) / 15.) Find the lateral area of a square pyramid with base edge 6in and height 4in. (LA = )
16.) Solve for x:
/ 17.) PQ = 16, OM = ______, Diameter = ______
/ 18.) CD=12, OM=8, ON=4,
EF = ___, radius = ___,
diameter = ___

19.) x = _____, y = _____
/ 20.) x = ____, y = _____
/ 21.) x = ____, y = ____,
z = _____

22.) x = _____
/ 23.) x = _____
/ 24.) x = _____

25.) x = ____
/ 26.) x = ____
/ 27.) Find the center and the radius of the circle: