1 / 17

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

nadine
Download Presentation

Chapter 6 Proportions and Similarity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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

  3. 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

  4. 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 4

  5. 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

  6. 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

  7. 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 ~

  8. 6-3 Similar Triangles (cont.) • Theorem 6.3 – similar triangles are reflexive, symmetric, and transitive SSS AA SAS

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

  10. 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

  11. 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

  12. Problem • Find x and ED if AE = 3, AB = 2, BC = 6, and ED = 2x - 3 Answer: x = 6 and ED = 9 12

  13. 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

  14. 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

  15. 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. 15

  16. Time Left?

  17. 6-6 Fractals!

More Related