1 check your homework assignment with your partner!

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Check your homework assignment with your partner!

2

13.1 Ratio & Proportion

The student will learn about:

ratios,

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similar triangles andproportions,

some special triangles.

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Ratios.

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

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Proportions.

A proportion is a statement that two ratios are equal. i.e.

a is the first termb is the second termc is the third termd is the fourth term

a c

b d

a and d are the extremes.

b and c are the means.

d is the fourth proportion.

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Proportions.

If

Then b is called the geometric mean between a and c and

a b

b c

b ac

Not to be confused with the arithmetic mean.

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Geometric Mean.

a bIf then b iscalled geometricmeanbetweenaandc.

b c

It is easy to show that b = √(ac)

Construction of the geometric mean.

or 6 = √(4 · 9)

4 636 36

6 9

CF = 9.00 cm

m CE = 5.99 cm

m CD = 3.99 cm

E

D FC

a c

b

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Theorems.

a cIf then ad bc. Multiply bothsidesby bd.

b d

a bIf ad bc then . Divide both sides by cd.

c d

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Theorems.

a c a b c dIf then . Add 1to bothsides.

b d b d

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

a c a b c dIf then . Subtract 1from bothsides.

b d b d

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.

TheoremBut 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.

A D

C

B1

h ADk ADC 21k DBarea C h DBDBC

area

D

A C D

B

A

2

D

h

Now to the proportionality theorem and its converse for our work on

similar triangles.

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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.

AB AC

AD AE

A

ED

CB

If a line parallel to one side of a triangle intersects the other two side, then it cuts off proportional segments.

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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?

Why?

Why?

Why?

QED

AD

CBE

TheoremWhy?

(6) k ∆BDE = k ∆CDE Same bases & altitudes.Why?

Why?3, 5 & 6.


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Previous slide.

Equals added

Substitution.

Why?

Why?

QED


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Previous slide.

Equals added

Substitution.

Why?

Why?

QED

(7) BD CE

AD AE

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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.

AB AC

AD AE

A

ED

CB

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.

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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?

Why?Why?

Why?

QED

(5) C= C’ Why?Prop of proportions

(6) → ← Why?Unique parallel assumed

A

D

CB

E

C’

Triangle Similarity

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Definition. If the corresponding angles in two triangles are congruent, and the sides are proportional, then the triangles are similar.

B

A

C

D

FE

AB AC BC

DE DF EF

Basic Similarity Theorems

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AAA Similarity

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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.

AB AC BC

DE DF EF

D

FE B

A

C

If the corresponding angles in two triangles are congruent, then the triangles are similar.

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Given: A=D, B=E, C=F

(1) E’ so that AE’ = DE Construction

(2) F’ so that AF’ = DF Construction

(3) ∆AE’F’ ≌ ∆DEF SAS.(4) AE’F =E = B CPCTE & Given


Why?

Why?Why?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: AB AC BC

DE DF EF

F’E’

B

A

C

D

FE

AA Similarity

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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

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Theorem. If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar.

D

FE B

A

C

If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar.

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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


Why?

Why?Why?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

FE

(6) A = A ReflexiveWhy?

SSS Similarity

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Theorem. If the corresponding sides are proportional, then the triangles are similar.

D

FE B

A

C

Proof for homework.

Right Triangle Similarity

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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.

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Ratios.

The ratio of the sides of a 30-60-90 triangle is1 : √3 : 2

a b c

1 23 30

60a

cb

c

c

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Ratios.

The ratio of the sides of a 45-45-90 triangle is1 : 1 : √2

a b c

1 1 2

45 a

c b = a

45

a

a

QUIZ

In trapezoid ABCD we have AB = AD. Prove that BD bisects ∠ ABC.

A D

CB

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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.

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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

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