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=Chapter 8 Similar Shapes=

Objectives:

 * Construct a dilation of a segment and a point by using a scale factor.
 * Construct a dilation of a closed plane figure.

Definitions:

 * 1) A **[|dilation]** is an example of a transformation that is not rigid. Dilations preserve the shape of an object, but they may change its size.
 * 2) The number //n// is called the **scale factor** of the transformation.
 * 3) In a dilation, each point and its image lie on a straight line that passes through a point known as the **[|center of dilation.]**
 * 4) You may have noticed that the size of images of segments that have been dilated varies according to the scale factor. If the size of a figure is reduced by a dilation, the dilation is called a **contraction**. If the size of a figure is enlarged by a dilation, the dilation is called an expansion. for a dilation with a scale factor of //n//.

=Chapter 8 Similar Shapes=

Objectives:

 * Define similar polygons.
 * Use Properties of proportions and scale factors to solve problems involving similar polygons.

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 number

Cross-Multiplication Property
If a/b=c/d and be and d dont=0, then ad=bc.

Reciprocal Property
If a/b=c/d and a,b,c, and d dont=0, then b/a d/c.

Exchange Property
If a/b=c/d and a,b,c and d dont=0, then a/c=b/d.

"Add-One" Property
If a/b=c/d and b and d dont=0, then a+b/b=c+b/d. =Chapter 8 Similar Shapes=

Objectives:

 * Develop the AA Triangle Similarity Postulate and the SSS and SAS Triangle Similarity Theorems.

Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Side-Side-Side Similarity Theorem
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Side-Angle-Side Similarity Theorem
If two sides of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. =Chapter 8 Similar Shapes=

Objectives:

 * Develop 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 Proportionality Corollary
Three or more parallel lines divide two intersecting transversal's proportionally. =Chapter 8 Similar Shapes=

Objectives:

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

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. =Chapter 8 Similar Shapes=

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.

Solids are similar if they are the same shape and all corresponding linear dimensions are proportional. For example, two rectangular prisms are similar if the lengths and widths of the corresponding faces and bases are proportional.

Cross-Sectional Areas,weight and Height
The amount of weight that a structure can support is proportional to its cross-sectional area. For example, column A, Whose radius is 3 times the radius of column B, can support 9 times more weight than column B. This is because the cross-sectional area of column A is 9 times that of column B.

(cross-section=3r) Column A (cross-section=r) Column B