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Check your homework assignment with your partner!

Check your homework assignment with your partner!. 1. 13.1 Ratio & Proportion. The student will learn about:. ratios,. proportions,. similar triangles and. some special triangles. 2. 2. Ratios. A ratio is the comparison of two numbers by division. i.e. a/b. 3. Proportions.

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Check your homework assignment with your partner!

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  1. Check your homework assignment with your partner! 1

  2. 13.1 Ratio & Proportion The student will learn about: ratios, proportions, similar triangles and some special triangles. 2 2

  3. Ratios. A ratio is the comparison of two numbers by division. i.e. a/b. 3

  4. Proportions. A proportion is a statement that two ratios are equal. i.e. a is the first term b is the second term c is the third term d is the fourth term a and d are the extremes. b and c are the means. d is the fourth proportion. 4

  5. Proportions. If Then b is called the geometric mean between a and c and Not to be confused with the arithmetic mean. 5

  6. Geometric Mean. It is easy to show that b = √(ac) b a c or 6 = √(4 ·9) Construction of the geometric mean. 6

  7. Theorems. 7

  8. Theorems. These are merely the most useful of the equations that may be derived from the definition of proportion; there are many others. 8

  9. NOTE We will need a proportionality theorem and its converse for our work on similar triangles.

  10. Theorem The two triangles have the same base and altitudes, the lines are parallel, so they have the same area. But first let’s look at the following relationship.

  11. Theorem But first let’s look at the following relationship. The two triangles have different bases and the same altitudes, the lines are parallel. What is the relationship of their areas? The ratio of the areas is the same as the ratio of the bases!

  12. THEOREM: Triangles that have the same altitudes have areas in proportion to their bases. C A D B h

  13. Now to the proportionality theorem and its converse for our work on similar triangles.

  14. Basic Proportionality Theorem. If a line parallel to one side of a triangle intersects the other two sides, then it cuts off segments which are proportional to these sides. A D E C B 14

  15. If a line parallel to one side of a triangle intersects the other two side, then it cuts off proportional segments. A E D B C What is given? Given: DE ∥ BC What will we prove? Prove: AB/AD = AC/AE Construction Why? (1) Construct BE and DC. (2) Alt ∆BDE = alt ∆ADE Bases & vertex. Why? Why? Theorem (4) Alt ∆ADE = alt ∆CDE Bases and vertex. Why? Theorem Why? Same bases & altitudes. Why? (6) k ∆BDE = k ∆CDE 3, 5 & 6. Why? QED 15

  16. If a line parallel to one side of a triangle intersects the other two side, then it cuts off proportional segments. Given: DE ∥ BC Prove: AB/AD = AC/AE Previous slide. Why? Equals added Why? Substitution. QED 16

  17. If a line parallel to one side of a triangle intersects the other two side, then it cuts off proportional segments. Given: DE ∥ BC Prove: AB/AD = AC/AE Previous slide. Why? Equals added Why? Substitution. QED (7) 17

  18. Converse of the Basic Proportionality Theorem. If a line intersects two sides of a triangle , and cuts off segments proportional to these two sides, the it is parallel to the third side. A D E C B 18

  19. If a line intersects two sides of a triangle , and cuts off segments proportional to these two sides, the it is parallel to the third side. A E D B C What will we prove? Given: AD/AB = AE/AC What is given? Prove: DE ∥ BC By contradiction Why? (1) Let BC’ be parallel. Why? Previous theorem (2) AD/AB = AE/AC’ Given Why? (3) AD/AB = AE/AC Why? Axiom (4) AE/AC = AE/AC’ (5) C= C’ Why? Prop of proportions (6) → ← Why? Unique parallel assumed QED C’ 19

  20. Triangle Similarity D E F Definition. If the corresponding angles in two triangles are congruent, and the sides are proportional, then the triangles are similar. A B C 20

  21. Basic Similarity Theorems 21

  22. AAA Similarity D E F Theorem. If the corresponding angles in two triangles are congruent, then the triangles are similar. A Since the angles are congruent we need to show the corresponding sides are in proportion. B C 22

  23. If the corresponding angles in two triangles are congruent, then the triangles are similar. Prove: D A E F C B Given: A=D, B=E, C=F What will we prove? What is given? Why? Construction (1) E’ so that AE’ = DE (2) F’ so that AF’ = DF Why? Construction SAS. Why? (3) ∆AE’F’ ≌ ∆DEF (4) AE’F =E =  B Why? CPCTE & Given (5) E’F’ ∥ BC Why? Corresponding angles Why? Prop Thm (6) AB/AE’ = AC /AF’ E’ F’ (7) AB/DE = AC /DF Why? Substitute (8) AC/DF = BC/EF is proven in the same way. QED 23

  24. AA Similarity A D E B F C Theorem. If two corresponding angles in two triangles are congruent, then the triangles are similar. In Euclidean geometry if you know two angles you know the third angle. 24

  25. SAS Similarity A D E F B C Theorem. If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar. 25

  26. If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar. A E’ F’ B D C E F What will we prove? What is given? Given: AB/DE =AC/DF, A=D Prove: ∆ABC ~ ∆DEF Why? Construction (1) AE’ = DE, AF’ = DF Why? SAS (2) ∆AE’F’ ≌ ∆DEF Why? Given & substitution (1) (3) AB/AE’ = AC/AF’ Basic Proportion Thm Why? (4) E’F’∥ BC (5) B =  AE’F’ Why? Corresponding angles (6) A =  A Reflexive Why? AA Why? (7) ∆ABC ≈ ∆AE’F’ (8) ∆ABC ≈ ∆DEF Why? Substitute 2 & 7 QED 26

  27. SSS Similarity Theorem. If the corresponding sides are proportional, then the triangles are similar. A D E F B C 27 Proof for homework.

  28. Right Triangle Similarity C b a h x c - x c A B Theorem. The altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle. 28 Proof for homework.

  29. Two Special Triangles.

  30. Ratios. 30 c c b a 60 c The ratio of the sides of a 30-60-90 triangle is 1 : √3 : 2 30

  31. Ratios. a 45 a c b = a 45 a The ratio of the sides of a 45-45-90 triangle is 1 : 1 : √2 31

  32. QUIZ In trapezoid ABCD we have AB = AD. Prove that BD bisects ∠ ABC. A D B C

  33. Summary. • We learned about ratios. • We learned about proportionality. • We learned about the geometric means. • We learned about the “Basic Proportionality Theorem” and its converse. 33

  34. Summary. • We learned about AAA similarity. • We learned about SAS similarity. • We learned about SSS similarity. • We learned about similarity in right triangles. 34

  35. Assignment: 13.1

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