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**Dilations**
Objectives: Vocab: - Dilation: a dilation is an example of a transformation that is not rigid. Dilations preserve the shape of an object, but they may change the size. - Scale factor: the number //n// is called the scale factor of the transformation. A great link to help you understand::http://regentsprep.org/regents/Math/codilate/Ldilate.htm
 * Dilations**
 * 1) Construct a dilation of a segment and a point by using a scale factor.
 * 2) Construct a dilation of a closed plane figure.

Question: What is different about a dilation from other transformations?



Objectives: Vocab: - //none// Blue Boxes -Definition: 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 Two polygons are similar if and only if there is a way of setting up a correspondence between their sides and angles such that the following conditions are met: ~ Each pair of corresponding angles is congruent. ~ Each pair of corresponding sides is proportional. -Properties of Proportions Let a, b, c, and d be any real numbers Cross-Multiplication Property If a/b=c/d and b and d do not equal 0, then ad=bc. Reciprocal Property If a/b=c/d and a, b, c, and d do not equal 0, then b/a=d/c Exchange Property If a/b=c/d and a, b, c, and d do not equal 0, then a/c=b/d “Add-One” Property If a/b=c/d and b and d do not equal 0, then a+b/b=c+d/d
 * Polygon Similarity**
 * 1) Define //similar polygons//
 * 2) Use properties of proportions and scale factors to solve problems involving similar polygons//.//

Question: Given the proportionality statement AB/ST=CD/UV=EF/WX=GH/YZ for two similar rectangles, write a similarity statemnt that displays the right correspondence.

A great link to help you understand::http://library.thinkquest.org/20991/geo/spoly.html



Objectives: Vocab: - //none// Blue Boxes: -AA (Angle-Angle) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. -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 similar. -SAS (Side-Angle-Side) Similarity Theorem If two of one triangle are proportional to two sides of another triangle and their angles are congruent, then the triangles are similar. A great link to help you understand::http://www.nos.org/Secmathcour/eng/ch-17.pdf
 * Similarity in Triangles**
 * 1) Develop the AA Triangle Similarity Postulate and the SSS and SAS Triangle Similarity Theorems.

Question: There are two triangles with sides equal to 6-6-12. Can you make a similarity statement about them? If so, what is it?



Objectives: Vocab: - //none// Blue Boxes: -Side-Splitting Theorem A line parallel 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.
 * Side – Splitting Theorem**
 * 1) Develop and prove the Side – Splitting Theorem.
 * 2) Use the Side – Splitting Theorem to solve problems.

Example: Here are some other proportions that can be found in a triangle by using the side-splitting theorem upper left/lower left=upper right/lower right upper left/upper right=lower left/lower right upper left/whole left=upper right/whole right lower left/whole left=lower right/whole right

A great link to help understand::http://www.phschool.com/atschool/ucsmp/geometry/Teacher_Area/GEO_TC13.html



Objectives: Vocab: - //none// Blue Boxes: - 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.
 * Indirect Measurement and Additional Similarity Theorems**
 * 1) Use triangle similarity to measure distances indirectly.
 * 2) Develop and use similarity theorems for altitudes and medians of triangles.

Example:One way to find the height of a basketball hoop would be to measure the shadow of the hoop, the height of a person, and the shadow of the person. You then use the proportion: Hoop/Person=Hoop shadow/Person shadow.

A great site to help you understand::https://www.icoachmath.com/sitemap/IndirectMeasurement.html



Objectives: Vocab: - //none// Blue Boxes: - //none//
 * Area and Volume Ratios**
 * 1) Develop and use ratios for areas of similar figures.
 * 2) Develop and use ratios for volumes of similar solids.
 * 3) Explore relationships between cross – sectional area, weight, and height.

Example: There are two squares. One has a side length of 4, and the other 1. The sides ratio would be 4/1. So their Area and volume ratio would be 16/1. Tada!

A great site to help you understand::[|http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html]