Ananth Dandibhotla, William Chen, Alden Ford, William Gulian Chapter 6 Proportions and Similarity

 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 Key Vocabulary

 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 6-1 Proportions Ratios

Bob made a 18 in. x 20 in. model of a famous painting. If the original painting’s dimensions are 3ft x a ft, find a. 4 Problem Answer: a = 10/4

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 proportional. 2 : 1 The order of the points matters

△ 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? 6 Problem Answer: They are not similar.

 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

 Theorem 6.3 – similar triangles are reflexive, symmetric, and transitive 6-3 Similar Triangles (cont.) SSS AA SAS

 Determine whether each pair of triangles is similar and if so how? 9 Problem Answer: They are similar by the SSS Similarity

 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

 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 Parallel Lines and Proportional Parts (Cont.)

Find x and ED if AE = 3, AB = 2, BC = 6, and ED = 2x Problem Answer: x = 6 and ED = 9

 Proportional Perimeters Thm. – If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides  Thm – 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 6-5 Parts of Similar Triangles

Find the perimeter of △ DEF if △ ABC ~ △ DEF, Ab = 5, BC = 6, AC = 7, and DE = Problem Answer: The perimeter is 10.8

» , Warsaw, Poland » A mathematician, Sierpiński studied in the Department of Mathematics and Physics, at the University of Warsaw in Graduating in 1904, he became a teacher of the subjects. » The Triangle: If you connect the midpoints of the sides of an equilateral triangle, it’ll 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 Sierpiński, in » Other Sierpiński fractals: Sierpiński Carpet, Sierpiński Curve » Other contributions: Sierpiński numbers, Axiom of Choice, Continuum hypothesis » Completely unrelated: There’s a crater on the moon named after him. 15 Wacław Sierpiński and his Triangle

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6-6 Fractals!