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=__Dilation's and Scale factors__=

Objectives:

 * By using a scale factor we will construct a dilation of a point and a segment.
 * Also construct a closed plane figure.

Vocabulary:

 * Dilation: A translated figure that is not rigid. The dilation in a coordinate plane is found by multiplying x and y by the same number (n).
 * D(x,y)=(nx,ny)
 * Scale Factor: The number that the points are multiplied by (n)
 * Center Of dilation: The center point of the dilation.
 * Contraction: When the figure's size is reduced by a dilation.
 * Expansion: When the figure's size is enlarged by the dilation.

__**Example 1:**__ The scale factor is 4. The image is (24,16). What is the pre-image? (For Answers Click here)

//(The center triangle is a contraction of the outer triangle)//

[[image:crflower15.jpg width="45" height="59" align="left"]]
=__Similar Polygons__=


 * Similar Polygons definition.
 * Using proportions and scale factors to solve for similar polygons.

Vocabulary:

 * Similar Figures: The two figures are congruent ONLY if one image is congruent to the other image by a dilation.
 * Dilation: An enlargement of an image that is a scale factor of the pre-image.

Polygon Similarity Postulate:
Two Polygons are similar if there is any corresponding angles or sides that are:
 * Their pairs of Corresponding angles are congruent.
 * Their pairs of Corresponding angles are congruent.

Properties of Proportions:

 * a,b,c and d are all real numbers
 * Cross Multiplication Property:**
 * When a/d=c/d and b and d DO NOT equal 0, that means ad=bc


 * Reciprocal Property:**
 * When a/b=c/d and a,b,c and d DO NOT equal 0, that means b/a=d/c


 * Exchange property:**
 * When a/b=c/d and a,b,c and d DO NOT equal 0, that means a/c=b/d


 * "Add One" Property:**
 * When a/b=c/d and b and d DO NOT equal 0, that means (a+b)/b=(c+d)/d

__**Example 2**__:

A triangle has sides xyz. x=3, y=5, and z=4. Another triangle has sides abc. a=6, b=10, and c=8. Are these triangles similar? If so why? (For answer click here.)

[|Picture link.]

[[image:crflower15.jpg width="50" height="52" align="left"]]
=__8.3 Triangle Similarity__=

Objective:

 * Help understand the AA Triangle Similarity Postulate and the sss and SAS Triangle Similarity Theorems

Vocabulary:

 * Similarity: Angles need to be congruent, sides need to be proportional.

AA ( Angle-Angle) Similarity Postulate:
If there are two angles in one triangle that happen to be congruent to two angles of another triangle, that means that they are similar.

SSS (Side-Side-Side) Similarity Theorem:
If all three sides of a triangle are proportional to all three sides of another triangle, that means the triangles are similar.

SAS (Side-Angle-Side) Similarity Theorem:
If two sides of one triangle are proportional to two sides of another triangle, and their included angles are congruent, then the triangles are similar.

__**Example 3:**__ Side AB of triangle ABC is 6, side BC is 9 and angle B is 50°. Side DE of triangle DEF is 45, side EF is 67.5, and angle E is 50°. Is triangle ABC is similar to triangle DEF? If so by which property? (AA, SSS, SAS).

(These triangles show the Angle Angle Similarity Postulate.)
=__8.4 The Side Splitting Theorem__= ===

Objectives:

 * //Develope and prove the side splitting theorem.//
 * //Use the side-splitting theorem to solve problems.//

//__Side Splitting Theorem__//
A line parallel to one side of the triangle divides the other two sides proportionally.

//__Two Transversal Proportionally Corollary__//
Three or more parallel lines divide two intersecting transversals proportionally.

=__8.5 Indirect Measurement and Additional Similarity Theorems__= ===

Objectives:

 * Use the triangle similarity to measure distances indirectly.
 * Develop and use similarity theorems for altitudes and medians of triangles.

//__Propotional 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 ats the other two sides.

=__8.6 Area and Volume Ratio__=

Objectives

 * Develop and use ratios for areas of similar figures.
 * Develop and use ratios for volumes of similar solids.
 * Explore relationships between cross-sectional area, weight, and height.

===A puppy that is 1ft tall at the shoulder has a leg bone whos circular cross section has a radius of 1cm. How much thicker would the leg bone need to be to give the same support to a dog that is 2ft tall at the shoulder? By what scale factor does the original radius need to be multiplied to provide a sufficient cross-sectional area?=== =(Answer52)=



[|Three Similar Polygons]
[|Directly Similar Figures]