Chapter 6 Proportions and Similarity Ananth Dandibhotla, William Chen, Alden Ford, William Gulian Key Vocabulary Proportion An equality statement with 2
ratios Cross Products a*d and b*c, in a/b = c/d Similar Polygons Polygons with the same shape Scale Factor A ratio comparing the sizes of similar polygons Midsegment A line segment connecting the midpoints of two sides of a triangle 6-1 Proportions Ratios compare two values, a/b, a:b (b 0)
For any numbers a and c and any non-zero number numbers b and d: a/b = c/d iff ad = bc Ratio s Problem Bob made a 18 in. x 20 in. model of a famous painting. If the original paintings dimensions are 3ft x a ft, find a.
Answer: a = 10/4 4 6-2 Similar Polygons Polygons with the same shape are similar polygons ~ means similar Scale factors compare the lengths of corresponding pieces of a polygon
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding angles are 2 : 1 proportional. The order of the points matters Problem
ABC and DEF have the same angle measures. Side AB is 2 units long Side BC is 10 units long Side DE is 3 units long Side FD is 15 units long Are the triangles similar? Answer: They are not similar. 6
6-3 Similar Triangles Identifying Similar Triangles: AA~ -Postulate- If the two angles of one triangle are congruent to two angles of another triangle, then the triangles are ~ SSS~ -Theorem- If the measures of the corresponding sides of two triangles are proportional, then the triangles are ~ SAS~ -Theorem- If the measures of two sides of a
triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, the triangles are ~ 6-3 Similar Triangles (cont.) Theorem 6.3 similar triangles are reflexive, symmetric, and transitive SSS
AA SAS Problem Determine whether each pair of triangles is similar and if so how? Answer: They are similar by the SSS Similarity
9 6-4 Parallel Lines and Proportional Parts Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other two sides in two distinct point, then it separates these sides into segments of proportional length Tri. Proportion Thm. Converse If a line
intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side 6-4 Parallel Lines and Proportional Parts (Cont.) Midsegment is a segment whose endpoints are the midpoints of 2 sides of a triangle. Midsegment Thm: A midsegment of a triagnle
is parallel to one side of the triangle , and its length is one- half the length of that side. Corollary 6.1: If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. Corollary 6.2: If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. 11
Problem Find x and ED if AE = 3, AB = 2, BC = 6, and ED = 2x - 3 Answer: x = 6 and ED =9 12 6-5 Parts of Similar Triangles
Proportional Perimeters Thm. If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides Thm 6.8-6.10 triangles have corresponding (altitudes/angle bisectors/medians) proportional to the corresponding sides Angle Bisector Thm. An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other
two sides Problem Find the perimeter of DEF if ABC ~ DEF, Ab = 5, BC = 6, AC = 7, and DE = 3. Answer: The perimeter is 10.8 14
Wacaw Sierpiski and his Triangle 1882-1969, Warsaw, Poland A mathematician, Sierpiski studied in the Department of Mathematics and Physics, at the University of Warsaw in 1899. Graduating in 1904, he became a teacher of the subjects. The Triangle: If you connect the midpoints of the sides of an equilateral triangle, itll form a smaller triangle. In the three triangular spaces, you can create more triangles by repeating the process, indefinitely. This example of a fractal (geometric figure created by iteration, or repeating the same procedure over and over again) was described by Sierpiski, in 1915. Other Sierpiski fractals: Sierpiski Carpet, Sierpiski Curve Other contributions: Sierpiski numbers, Axiom of Choice, Continuum hypothesis
Completely unrelated: Theres a crater on the moon named after him. 15 Time Left? 6-6 Fractals!