1 Check your homework assignment with your partner!

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

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

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.

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

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

7 Theorems.

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

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

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

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!

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

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 E D C B

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

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

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

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 E D C B

19 Given: AD/AB = AE/AC Prove: DE ∥ BC (1) Let BC’ be parallel. By contradiction (2) AD/AB = AE/AC’ Previous theorem (3) AD/AB = AE/AC Given (4) AE/AC = AE/AC’ Axiom What is given? What will we prove? Why? QED (5) C= C’ Why?Prop of proportions (6) → ← Why?Unique parallel assumed A D C B E C’

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

Basic Similarity Theorems 21

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

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

AA Similarity 24 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. F D E B A C

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

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

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

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

Two Special Triangles.

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

31 Ratios. The ratio of the sides of a triangle is 1 : 1 : √2 45a cb = a 45 a a

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

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

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

Assignment: 13.1