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=CHAPTER 8.1= OBJECTIVES- Construct a dilation of a segment and a point by using a scale factor. OBJECTIVE- Construct a dilation of a closed plane figure.

DEFINITIONS!
--Dilation- An example of a transformation that is not rigid. They preserve the shape of an ovject, but they may change in size (making it larger or smaller) - You may find this by multiplying X and Y coordinates of a point by the same number - **D(X,Y)=(nX,nY)** D-//Dilation// n-//Same number// --Scale Factor- The number by which the distance of the preimage from the center of dilation is multiplied to dertermine the distance of the image point from the center. --Conjecture- The distance from the origin to the image of a point transtormed by a dialation with a scale factor N is N timesthe distance from the origin to the preimage. --Contraction- A constraction asserts that a statement and its negatio nare both true. //(tighten, smaller, shrink-age)// --Expansion- A dialation in which the preimage is enlarged in size. //(bigger, expand)//


 * n|<1 - CONTRACTION
 * n|>1 - EXPANSION



Example: What is the image of the point (4, 6) transformed by the dilation D(x, y)= (12x, 12y)? What's the scale factor?

Solution: The image is the point D(4, 6)= (12 x 4, 12 x 3) = (48, 36). The scale factor is the multiplier, 12.

=CHAPTER 8.2= OBJECTIVES- Define similar polygons. OBJECTIVES- Use Properties of Proportions and Scale Factors to solve problems involving similar polygons.

Two figures are similar //if and only if// one is congruent to the image of the other by a dialation
 * WHAT ARE SIMILAR FIGURES?!** __...well, i'm glad you asked!__

PROPERTIES AND POSTULATES-
__Polygon Similarity Postulate__- 2 polygons are similar if and only if there is a way of setting up a corrispondence between sides and angles such that the following conditions are met... - each pair of correspoinding angles is congruent - each pair of correspoinding sides is proportional

Cross Multiplication Property- if a/b equals c/d and b and d don't equal 0, then ad equals bc. Reciprocal Property- If a/b equals c/d and a,b,c, and d don't equal 0, then b/a equals d/c. Exchange Property- If a/b equals c/d and a,b,c, and d don't equal 0, then a/c equals b/d. Add One Property- If a/b equals c/d and d doesn't equal 0, then a+b/b equals c+d/d.

Example: Two people are competing to see who can make the larger building. After they are finished, they begin comparing. It is no contest that one of their buildings is extremely larger than the other, but they decide the measure anyway. The measurements of the larger building are, in meters, 30x60. The measurements of the smaller building cannot be completely found because a giant fire is consuming the shorter side of the building. The longer side's measurement is 20 meters. Find the shorter side, 'n'.

Solution: n/20 equals 30/60 20 x n/20 equals 30/60 x 20 n equals 600/1200 meters.

=CHAPTER 8.3= OBJECTIVE- Develop the AA Triangle Similarity Postulate and the SSS and SAS Triangle Similarty Theorems.

POSTULATES AND THEOREMS-
--AA(Angle-Angle) Similarity Postulate- If 2 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 3 sides of one triangle are proportional to the 3 sides of another triangle, then the triangles are similar. --SAS (Side-Angle-Side) Similarity Theorem- If 2 sides of one triangle are proportional to 2 sides of another triangle and their included angles are congruent then the angles are similar.

Example: Imagine that there are two triangles that you have to find if they are simlar for a math problem. Right Triangle ABC has the C angle equaling 35 degrees, while Right Triangle DEF has the D angle equaling 55 degrees. Are they similar?

Solution: By the triangle sum theorem, the measurement of angle A equals 180 degrees - 90 degrees - 35 degrees 55 degrees. So, measurement of angle A equals measurement of angle P (A is congruent to P), and measurement of angle B equals measurement of angle E (B is congruent to E). Thus, by the AA similarity postulate, triangle ABC is similar to triangle DEF.

=CHAPTER 8.4= OBJECTIVES- Develop and prove the Side-Splitting Theorem. OBJECTIVES- Use the Side-Splitting Theorem to solve problems.

A line parallel to one side of the triangle divider the other two sides proportionally.
 * WHAT IS THE SIDE SPLITTING THEOREM?** __... I'm glad you asked__

TWO TRANSVERSAL PROPORTIONALITY COROLLARY- Three or more parallel lines divid two intersecting transversals proportionally.

Example: Remember that upper left/lower left equals upper right/lower right.

Solution: 1. 12/16 equals 15/x 2. (cross multiply) 12x equals 240 3. x equals 20

=CHAPTER 8.5= OBJECTIVES- Use triangle s imilarity to meauser distances in directly. OBJECTIVES- Develop and use similarity theorms for altitudes and medians of triangles.

THEORMS!
__Proportional Altitudes Theorem-__ If 2 triangles are similar, then their corresponding altitudes have the same ratio as their corresponding sides. __Proportional Medians Theorem-__ If 2 triangles are similar, then their corresponding medians have the same radio as their corresponding sides. __Proportional Angle Bisecors Theorem-__ If 2 triangles are similar, then their corresponding angle bisectors have the same ratio as the corresponding sides. __Proportional Segments Theroem-__ An angle bisector of a triangle divides the opposite side into 2 segments that have the same radio as the other 2 sides.

Example: The upper left length is 10, and the lower length is 12. The upper right length is X, and the lower right length is 14. Find X.

Solution: 1. 10/12 equals x/14 2. (cross multiply) 140 equals 12x 3. x equals 11.66667

=CHAPTER 8.6= OBJECTIVES- Develop and use ratios for areas of similar figures. OBJECTIVES- Develop and use ratios for volumes of similar solids. OBJECTIVES- Explore relationships between cross-sectional area, weight and height.

Example: Imagine that you have an extremely large arm that cannot stop growing. The height of your arm increases by a factor of 2, and the volume of it increases by a factor of 4. You need an arm bone with the cross-sectional area of 4 times larger than that of your original arm size. Thus, 4 x pi x r^2 is the required cross-sectional area ('r' being radius). In the following equation, R will represent the radius of the arm bone of your larger arm:



Solution:

pi x r^2 = 4 x pi^2 R^2 = 4 x r^2 R= square root of 4 x r The radius of the bone must be the square root of 4.