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CHAPTER 10 Similarity

CHAPTER 10 Similarity. Melanie Wagner. LESSON 1. DEFINTIONS ratio- the ratio of the number a to the number b is the number a/b Proportion- a proportion is an equality between two ratios Example: a/b = c/d ad = bc

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CHAPTER 10 Similarity

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  1. CHAPTER 10Similarity Melanie Wagner

  2. LESSON 1 DEFINTIONS • ratio- the ratio of the number a to the number b is the number a/b • Proportion- a proportion is an equality between two ratios Example: a/b = c/d ad = bc • Geometric mean- the number b is the geometric mean between the numbers a and c if a, b and c are positive and a/b = b/c

  3. LESSON 2 • A correspondence between their (figures that are similar) points such that corresponding segments are proportional is a similarity DEFINITIONS • Similar- Two triangles are similar iff there is a correspondence between their vertices such that their corresponding sides are proportional and their corresponding angles are proporional • ABC ~ DEF

  4. LESSON 2 • A key to understanding similarity is the dilation transformation • Center of dilation: point O = center of dilation • GEF ~ G’E’F’ • Magnitude: gives the relative size of the image compared with the original (ratio of the corresponding lengths) ex. OE = rOE’, OG= rOG’, and OF= rOF’

  5. LESSON 3 • The Side-Splitter Theorem- if a line parallel to one side of a triangle intersects the other two sides in different points, it divides the sides in the same ratio • BX/XA = BY/YC * BX/BY=XA/YC why?

  6. LESSON 3 • Corollary to the Side-Splitter Theorem- if a line is parallel to one side of a triangle and intersects the other two sides in different points, it cuts off segments proportional to the sides • jhfkdhsaklhfkd;ahfklas;BX/BA = BY/BC and AX/AB = CY/CB • *Midsegment Theorem: parallel to third side and half as long

  7. LESSON 4 • The AA Theorem: if two angles of one triangle are equal to two angles of another triangle, the triangles are similar *Only 2 angles are required because…

  8. Corollary to the AA theorem: two triangles similar to a third triangle are similar to each other

  9. LESSON 5 • Corresponding altitudes: Altitudes in two triangles drawn from corresponding vertices of the triangles • Theorem 46: Corresponding altitudes of similar triangles have the same ratio as that of the corresponding sides

  10. LESSON 6 • Theorem 47: The ratio of the perimeters of 2 similar polygons is equal to the ratio of the corresponding sides • Theorem 48: The ratio of the areas of 2 similar polygons is equal to the square of the ratio of the corresponding sides Theorem 47 4 8/16 2/4 2 2/4=2/4 4 Theorem 48 2 4/16= 22/42 INSIGHT* you may be asked to find the ratio of the areas when given the ratio of perimeter

  11. EXTRA LESSON • Angle Bisector Theorem: An angle bisector in a triangle divides the opposite side into segments that have the same ratio as the other two sides CD/DB=CA/AB *If CB is x and CD=3, AC= 5, and AB=8…

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