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.

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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 longSide BC is 10 units longSide DE is 3 units longSide FD is 15 units long

Are the triangles similar?

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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.)

SSSAA

SAS

Determine whether each pair of triangles is similar and if so how?

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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. 11

6-4 Parallel Lines and Proportional Parts (Cont.)

Find x and ED if AE = 3, AB = 2, BC = 6, and ED = 2x - 3

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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 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

6-5 Parts of Similar Triangles

Find the perimeter of △DEF if △ABC ~ △DEF, Ab = 5, BC = 6, AC = 7, and DE =

3.

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Problem

Answer: The perimeter is 10.8

» 1882-1969, Warsaw, Poland» A mathematician, Sierpiński 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, 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 1915.

» 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.

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Wacław Sierpiński and his TriangleWacław Sierpiński and his Triangle

Time Left?

6-6 Fractals!