schugi2thacas2

=Dilations and Scale Factors=

[[image:http://farm1.static.flickr.com/22/33358201_f597b73113_b.jpg width="163" height="121" link="http://www.flickr.com/photos/bitterlysweet/33358201/"]]
//Objective:// Constuct a dilation of a segment ans a point by using a scale factor. //Objective//: Consrtuct a dilation of a closed plane figure.


 * Dilation:** Example of a transformation that isn't rigid.
 * Center of Dilation:**Each point ad its image lie on a straight line tat passes through a point.
 * Contraction:** The size of the figure is reduced by a dilation.
 * Expansion:** The size of a figure is enlarged by a dilation.



__The bigger triangle is a dilation of the smaller triangles, because it retains the original shape as well as the shapes original angles.

[|unserstanding dilation webpage]__

= = =Similar Polygons=

[[image:http://farm1.static.flickr.com/22/33358201_f597b73113_b.jpg width="163" height="121" link="http://www.flickr.com/photos/bitterlysweet/33358201/"]]
//Objective:// Define similar polygons. //Obective:// Use properties of proportions and scale factors to solve problems involving similar polygons.

Each pair of corresponding sides is proportional.
 * Similar Figures:** Two figures are similar if and only if one is congruent to the image of the other by a dilation.
 * Polygon Similarity Postulate:** Each pair of corresponding angles is congruent.

Properties of proportions
If a/b=c/d and b and d don't equal 0, then ad=bc If a/b=c/d and a, b, c and d don't equal 0, then b/a=d/c If a/b=c/d and a, b, c and d don't equal 0, then a/c=b/d If a/b=c/d and b and d don't equal 0, then a+b/b=c+d/d
 * Cross Multiplication Property**
 * Reciprocal Property**:
 * Exchange Property**:
 * Add One Property**:

=Triangle Similarity=

[[image:http://farm1.static.flickr.com/22/33358201_f597b73113_b.jpg width="163" height="121"]]
//Objective:// Develop the AA triangle Similarity Postulate and the SSS and SAS Triangle SImilarity Theorems.

to two angles of another triangle, then the triangles are similar. proportional to the three sides of another triangle, then the triangles are similar. proportional to two sides of another triangle and their included angels are congruent, then the triangles are similar.
 * AA Similarity Postulate:** If two angles of one triangle are congruent
 * SSS Similarity Theorem:** If the three sides of one triangle are
 * SAS Similarity Theorem:** If two sides of one triangle are



= = =The Side-Splitting Theorem=

[[image:http://farm1.static.flickr.com/22/33358201_f597b73113_b.jpg width="163" height="121" link="http://www.flickr.com/photos/bitterlysweet/33358201/"]]
//Objective:// Develop and prove the Side-Splitting Theorem. //Objective:// Use the Side-Splitting Theorem to solve problems.

__**Side-Splitting Theorem**__ A line on one side of a triangle that divides the other two proportionally.

Three or more parallel lines that divide into two intersecting transversals
 * __Two-Transversal Proportionality Corollary__**

= = =Indirect Measurement and Additional Similarity Theorems=

[[image:http://farm1.static.flickr.com/22/33358201_f597b73113_b.jpg width="163" height="121" link="http://www.flickr.com/photos/bitterlysweet/33358201/"]]
//Objective://Use triangle similarity to measure distances indirectly //Objective://Develop and use similariy theorems for altitudes and medians of triangles

If two triangles are similar, then the corresponding altitude have the same ratio as their corresponding sides
 * __Proportional Altitudes Theorem__**

If two triangles are similar the their corresponding medians have the same ratio as their corresponding sides
 * __Proportional Medians Theorem__**

If two trianlges are similar then their corresponding angle bisectors have the same ratio as the corresponding sides
 * __Proportional Angle Bisectors Theorem__**

__**Proportional Segments Theorem**__ An angle bisector of a triangle divides the opposite into two segments that have the same ratio as the other two sides

=Area and Volume Ratios=

[[image:http://farm1.static.flickr.com/22/33358201_f597b73113_b.jpg width="163" height="121" link="http://www.flickr.com/photos/bitterlysweet/33358201/"]]
//Objective//:Develope and use ratios for areas of similar figures //Objective//:Develope and use ratios for volume of similar solids //Objective://Explore relationships between cross-sectional area, weight, and height

[|area to volume ratios]