lesson 7-1: using proportions 1 ch 7 similarity. lesson 5-1: using proportions 2 lesson 7-1 using...

Lesson 7-1: Using Proportions 1

Ch 7

Similarity


Lesson 7-1

Using Proportions


A ratio is a comparison of two numbers such as a : b.

A

BC

D

3.6

6

8

4.8

10Now try to reduce the fraction.

Ratio:When writing a ratio, always express it in simplest form.

Example:

Ratio

?What is the ratio of AB to CB10

6

AB

CB

5:3.ratio of AB to CB

10 5

6 3


The baseball player’s batting average is 0.307 which means he is getting approxiamately one hit every three times at bat.

A baseball player goes to bat 348 times and gets 107 hits. What is the players batting average?

Solution:Set up a ratio that compares the number of hits to the number of times he goes to bat.

Convert this fraction to a decimal rounded to three decimal places.

Example……….

Ratio: 107

348

Decimal: 1070.307

348


Proportion: An equation that states that two ratios are equal.

Terms

Proportion

a c

b d

First Term

Second Term

Third Term

Fourth Term

Extremes: a and d

To solve a proportion, cross multiply the proportion:

Means : b and c

a d c b


Proportions- examples….Find the value of x.Example 1:

84 yards

2 ft

x

356 yards

Solve the proportion.Example 2:8x = 30

8 • x = 6 • 58x = 30 8 8

x = 3.755

6 8

x

(356 3) 1068

2 (84 3) 252

252 2136

21368.5

252

x

x

x ft

Lesson 5-2: Similar Polygons 7

Lesson 7-2

Similar Polygons


Similar PolygonsDefinition: Two polygons are similar if:

1. Corresponding angles are congruent.2. Corresponding sides are in proportion.

The scale factor is the ratio between a pair of corresponding sides.

Scale Factor:

Two polygons are similar if they have the same shape not necessarily have the same size.


Naming Similar PolygonsWhen naming similar polygons, the vertices (angles, sides) must be named in the corresponding order.

; ; ;

If ABCD PQRS

A P B Q C R D S

AB BC CD AD

PQ QR RS PS

A B

CD

P Q

RS


Example-

The two polygons are similar. Solve for x, y and z.

Step 3: Find the scale factor between the two polygons.

15 2 2 30 2

1 5 17.5 10

115

y

x zx y z

AD DC BC AB

EH HG FG EF

Note: The scale factor has the larger quadrilateral in the numerator and the smaller quadrilateral in the denominator.

Step1: Write the proportion of the sides.

Step 2: Replace the proportion with values.15 30 20

5 10

y

x z

Step 4: Write separate proportions for each missing side and solve.

5

z

x

10y

15

30

20

B C

AD

F G

E

H


Scale factor is same as the ratio of the sides. Always put the first polygon mentioned in the numerator.

Example:

What is the scale factor from ZYX to ABC?

The scale factor from ABC to ZYX is 2/1.

AB 18 2

ZY 9 1

If ABC ~ ZYX, find the scale factor from ABC to ZYX.

7 9

5

18

10

14

C

A

B

Z

YX

½

Lesson 5-3: Proving Triangles Similar

12

Lesson 7-3

(AA, SSS, SAS)

Proving Triangles Similar


13

AA Similarity (Angle-Angle)

A D

If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.

E

DA

B

CF

B E

ABC ~ DEFConclusion:

andGiven:


14

SSS Similarity (Side-Side-Side)If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar.

E

DA

B

CF

Given:

Conclusion:

5

11 22

8 1610

BC

EF

AB

DE

AC

DF

8

16

5

10

11

22

ABC ~ DEF


15

SAS Similarity (Side-Angle-Side)

ABC ~ DEF

If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar.

Given:

Conclusion:

E

DA

B

CF

5

11 22

10

AB ACA D and

DE DF


16

Similarity is reflexive, symmetric, and transitive.

1. Mark the Given.2. Mark …

Shared Angles or Vertical Angles3. Choose a Method. (AA, SSS , SAS)Think about what you need for the chosen method and be sure to include those parts in the proof.

Steps for proving triangles similar:

Proving Triangles Similar


17

Problem #1:

Pr :

Given DE FG

ove DEC FGC

CD

E

G

F

Step 1: Mark the given … and what it implies

Step 2: Mark the vertical angles

Step 3: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasons

Step 5: Is there more? Statements Reasons

Given

Alternate Interior <s

AA Similarity

Alternate Interior <s

1. DE FG2. D F 3. E G

4. DEC FGC

AA


18

Problem #2


Step 2: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasons

Step 5: Is there more? Statements Reasons

Given

Division Property

SSS Similarity

Substitution

SSS

: 3 3 3

Pr :

Given IJ LN JK NP IK LP

ove IJK LNP

N

L

P

I

J K

1. IJ = 3LN ; JK = 3NP ; IK = 3LP

2. IJ

LN=3,

JK

NP=3,

IK

LP=3

3. IJ

LN=

JK

NP=

IK

LP

4. IJK~ LNP


19

Problem #3


Step 3: Choose a method: (AA,SSS,SAS)

Step 4: List the Parts in the order of the method with reasons

Next Slide………….

Step 5: Is there more?

SAS

: int

int

Pr :

Given G is the midpo of ED

H is the midpo of EF

ove EGH EDF

E

DF

G H

Step 2: Mark the reflexive angles


20

Statements Reasons

1. G is the Midpoint of

H is the Midpoint of

Given

2. EG = DG and EH = HF Def. of Midpoint

3. ED = EG + GD and EF = EH + HF Segment Addition Post.

4. ED = 2 EG and EF = 2 EH Substitution

Division Property

Substitution

Reflexive Property

SAS Postulate

7. GEHDEF

8. EGH~ EDF

6. ED

EG=

EF

EH

5. ED

EG=2 and

EF

EH =2

ED

EF

Lesson 5-4: Proportional Parts 21

Proportional Parts

Lesson 7-4


Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

A B

C

D E

F

AB AC BC

DE DF EF

Similar Polygons


, CB CD

If BD AE thenBA DE

If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional length.

Side Splitter Theorem

1 2

34A

B

C

D

E

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.

, CB CD

If then BD AEBA DE


6 9

4 x

4x + 3

9

A

B

C

DE

2x + 3

5

2 3 4 3

5 95(4 3) 9(2 3)

20 15 18 27

2 12

6

x x

x x

x x

x

x

A

B

C

D E

If BE = 6, EA = 4, and BD = 9, find DC.

6x = 36 x = 6

Solve for x.

Example 1:

Example 2:

Examples………

6

4

9

x


Midsegment Theorem

A segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side.

R

S T

ML

int

int

1.

2

If L is the midpo of RS and

M is the midpo of RT then

LM ST and ML ST


, , , .AB DE AC BC AC DF

etcBC EF DF EF BC EF

If three or more parallel lines have two transversals, they cut off the transversals proportionally.

AB

C

D

EF

Extension of Side Splitter

If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.


Forgotten Theorem

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

sec ,AD AC

If CD is the bi tor of ACB thenDB BC

C

A

BD


(1) then the perimeters are proportional to the measures of the corresponding sides.(2) then the measures of the corresponding altitudes are proportional to the measure of the corresponding sides..(3) then the measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides..

B C

A

E F

D

HG I J

If two triangles are similar:

( )

( )

( sec )

( sec )

AG

D

Perimeter of ABC

Perimeter of DEF

altitudeof ABC

altitudeof DEF

anglebi tor of ABC

I

AH

DJ anglebi tor

AB BC AC

DE EF DF

of DEF

~ABC DEF


A

B

C

D

E

F

20 60

420 240

12

AC Perimeter of ABC

DF Perimeter of DEF

xx

x

The perimeter of ΔABC is 15 + 20 + 25 = 60.Side DF corresponds to side AC, so we can set up a proportion as:

Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4. Find the perimeter of ΔDEF.

Example:

15

20

254

lesson 7-1: using proportions 1 ch 7 similarity. lesson 5-1: using proportions 2 lesson 7-1 using...

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