Sections 1 *Dilations and Scale Factors*



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^^^Each shape is preserved but the size is changed by dilation^^^

Objectives:
  • Construct a dilation of a segment and a point by using a scale factor.
  • Construct a dilation of a closed plane figure.

Words and Formulas you need to know:


Words:
Dilation-
An example of a transformation that is not rigid. Dilations perserve the shape of the figure, but they may change its size.
Scale Factor- The number you multiply the -x and -y axis by of the transformation.
Center of dilation- In a dilation, each point and its image lie on a straight line that passes through this point.
Contraction- If the size of the figure is reduced because of the dilation it is called this.
Expansion- If the size is reduced because of the dilation it is then called this.

Quick Example 1:
If you have a image with the beginning cordinates:
(36,14) remember: (x,y)
and you had a scale factor of 2...
What would the cordinates of your pre-image be?

*see answer at bottom of page =]




Section 2 *Similar Polygons*


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^^^ Each shape or scale are similar polygons but different proportions ^^^

Objectives:
  • Define similar polygons.
  • Use Properties of Proportions and scale factors to solve problems involving similar polygons.

Words and Formulas you need to know:



Words:
Similar Figures-
Two figures are similar if and only one is congruent to the image of the other by the dilation.
Polygon Similarity Postulate- Two polygons are similar if and only if there is a way of setting up a correspondence between their sides and angles. So that these conditions are true fothe r those polygons:
  • Each pair of corresponding angles is congruent.
  • Each pair of corresponding sides is proportional.

Properties of Proportions:

If a, b, c, and d, were any real numbers...
  • Cross-Multiplication Property:
If a/b = c/d and b and d was not equal to 0...
then, ad = bc.
  • Reciprocal Property
If a/b = c/d and a, b, c, and d, was not equal to 0...
then, b/a = d/c
  • Exchange Property:
If a/d = c/d and a, b, c, and d, was not equal to 0..
then, b/a = d/c
  • "Add-One" Property:
If a/b = c/d and b, and d, was not equal to 0...
then, a+b/b = c+d/d

Quick Example 2:
If you had two triangles, The first triangle being abc:
a=12
b=18
c=54
The second triangle being def:
d=2
e=3
f=9
Are these two triangles similar? If so, how?

*see answer at bottom of page =]


Section 3 *Triangle Similarity*


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^^^ Each triangle shown is similar due to either the postulates of AA, SSS, or SAS^^^

Objectives:
  • Develope the AA Triangle Similarity Postulate and the SSS and SAS Triangle Similarity Theorems.

Postulates you will need to know:



  1. AA [Angle-Angle] Similarity Theorem- If two angles of one triangle are congruent to two angles of another triangle, then the triangles are congruent.
  2. SSS [ Side- Side- Side ] Similarity Theorem- If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are congruent.
  3. SAS [Side- Angle- Side] Similarity Theorem-SAS [Side- Angle- Side] Similarity Theorem- If two sides of one triangle are proportional to two sides of another triangle and their sides angles are congruent, then the triangles are congruent.

Quick Example 3:
If I had two similar triangles, abc and def..
ab=66
bc=94
ca=22
and triangle def...
de=8.25
bc=11.75
cd=2.75
Which postulate is being used to prove these are similar?? [ps. similarity theorems above ^^]

*see answer at bottom of page =]



Section 4 *The Side-Splitting Theorem*


picture #1
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^^^ Each triangle shown is cut into segments which can be measured by proportions theorems ^^^


Objectives:
  • Develop and prove the Side -Splitting Theorem.
  • Use the Side-Splitting Theorem to solve problems.

Words you need to know:



Words:
Side-Splitting Theorem-
A line parellel to one side of the triangle divides the other two sides proportionally.
Two-Transversal Proportionality Corollary- Three or more parallel lines divide two intersecting transversals proportionally.

Quick Example 4:
^^ If you had a triangle that was similar to picture #1 and was cut by one segment...
say the first section at the top of each side:
top left side=12
top right side=38.4
and the second section, cut by the segment, on the bottom of each side:
bottom left side= x
bottom right side=16

Solve for x, using the side-splitting theorem...

*see answer at bottom of page =]



Section 5 *Indirect Measurements and Additional Similarity Theorems*



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^^^ If you form a right triangle using the top of the object to the end of the shadow you will be able to measure distances indirectly^^^

Objectives:
  • Use triangle similarity to measure distances indirectly.
  • Develop and use similarity theorems for altitudes and medians of triangles.

Words you need to know:


Words:
Proportional Altitudes Theorem-
If two triangles are similar, then their corresponding altitudes have the same ratio as their corresponding sides.
Proportional Medians Theorem- If two triangles are similar, then their corresponding medians have the same ratio as their corresponding sides.
Proportional Angle Bisectors Theorem- If two triangles are similar, then their corresponding angle bisectors have the same ratio as the corresponding sides.
Proportional Segments Theorem- An angle bisector of a triangle divides the opposite side into two segments that have the same ratio as the other two sides.




Section 6*Area and Volume Ratios*


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^^^Which one would be able to hold the most water? Hold the most grain???^^^
*Find out using relationships between cross-sectional area, weight and height*

Objectives:
  • Develope and use ratios for areas of similar figures.
  • Develope and use ratios for volumes of similar solids
  • Explore relationships between cross- sectional area, weight, and height



Now for the answers.....

Quick Example 1:
To find the pre-image you would look at the starting cordinates.. (36,14)
First you want to divide your -x axis [36] by your scale factor [2]...36/2=18
Second, you want to do the same with your -y axis and divide by 2...14/2=7
So, look at your answers and the first one would become your pre-image -x axis number and the second, your -y axis number.
Your final answer for you pre-image cordinates would be: (18,7)

Quick Example 2:
The answer is yes, these two triangles are similar....
If you took each point of one triangle that correspondce with the point on the second triangle...ex. A corresponds with D and b corresponds with E and so on.... you would divide those numbers by eachother.
Like this:
a/d...12/2=6
b/e...18/3=6
c/f...54/9=6

You would see that all equations equal 6, which means you then have a dilation of 6, and two congruent triangles.

Quick Exmple 3:
This is one of the easier ones...what we need to do for this one is to take one side and divide it by the corresponding side..for example ab/de...bc/ef...like that.
66/8.25=8
94/11.75=8
22/2.75=8
You then see, by all the equations equaling 8, that this triangle is similar by SSS [side-side-side] because all three sides of the first triangle are shown to be congruent to all three sides of the second triangle.

Quick Example 4:
Key:
tl=top left
tr=top right
bl=bottom left
br=bottom right

12/5=38.4/x you would then have to cross-multilpy, which would give you...
12·x=5·38.4 which would then give you...
12x=192 You now need to get x by itself so you would now...
192/12 which gives you what x equals..
x=16
now you need to put your answer as a proportion to see if it works
12/5=38.4/16 which they both come out to equa...
2.4!!
and you're done.

And thats all I've got for you...
If you still need more help check out these websites below.


Use the following sites to get help and see examples:
1. http://www.freemathhelp.com/Lessons/Geometry_Similar_Triangles_BB.htm -- For help with similar triangles.
2. http://education.yahoo.com/homework_help/math_help/problem_list?id=minigeogt_8_1 --For practice problems and help with dilations and scale factors.
3. http://standards.nctm.org/document/eexamples/chap7/7.3/index.htm --For interactive help with area ratios.