Download Chapter 6 Proportions and Similarity

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
Ratios
Problem
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.
Answer: a = 10/4
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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
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.
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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
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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.
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Problem
Find x and ED if AE = 3, AB = 2, BC = 6, and ED =
2x - 3
Answer: x = 6 and ED = 9
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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
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Wacław Sierpiński and his Triangle
» 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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Time Left?
6-6 Fractals!