Download - Ananth Dandibhotla, William Chen, Alden Ford, William Gulian Chapter 6 Proportions and Similarity
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 longSide BC is 10 units longSide DE is 3 units longSide 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.)
SSSAA
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. 11
6-4 Parallel Lines and Proportional Parts (Cont.)
Find x and ED if AE = 3, AB = 2, BC = 6, and ED = 2x - 3
12
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.
14
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.
15
Wacław Sierpiński and his TriangleWacław Sierpiński and his Triangle
Time Left?
6-6 Fractals!