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14.1 Ratio & Proportion

The student will learn about:

1

similar triangles.

Triangle Similarity

2

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

AAA Similarity

3

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.

4

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

What is given? What will we prove?

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

E’

B

A

C

F’

AA Similarity

5

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

Theorem

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If a line parallel to one side of a triangle intersects the other two sides, then it cuts off a similar triangle. Don’t confuse this theorem with If a line intersects two sides of a triangle , and cuts off segments proportional to these two sides, then it is parallel to the third side.

CB

A

ED

Proof for homework.

SAS Similarity

7

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.

8

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?

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

10

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

Pythagoras Revisited

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From the warm up:

A

C

B

b a

c

h

c - x x

2short side a h xa cx

hypotenuse c b a

2long side b c x hb c(c x)

hypotenuse c b a

And of course then,

a 2 + b 2 = cx + c(c – x) = cx + c 2 – cx = c2

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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)

CF = 9.00 cm

m CE = 5.99 cm

m CD = 3.99 cm

E

D FC

4 636 36

6 9

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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: 14.1