Denton Independent School District / Overview



Name_________________________________________________________ Date_____________ Hour______

9.1 – Using Ratios and Proportions (

A _______________________ is a comparison of two quantities.

The ratio of a to b can be expressed as or or

Examples:

Write each ratio in simplest form-

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. six days to two weeks 6. 24 inches : 3 feet

7. 45 centimeters to 7 meters 8. 17 yards to 15 feet

9. 280 seconds : 6 minutes 10. 75 meters to 5 kilometers

A _______________________ is an equation that shows two equivalent fractions.

There are three methods to determine if a ratio forms a proportion.

Method 1 Method 2 Method 3

Simplify the fractions Determine the decimals Cross Multiply

[pic] [pic] [pic]

So, the answer is “YES” since the fractions, the decimals, and the cross product are equal.

Examples:

Determine whether the following are a proportion:

11. [pic] 12. [pic]

In the proportion below there are two cross-products.

11 and x _____________

[pic]

16 and 44 ____________

You can use cross-multiplication to solve equations in proportion form…

Examples:

Solve each proportion by using cross-products.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic]

Geometry G Name ________________________

Ratios Worksheet 1

Express each ratio in lowest terms.

1. 8 to 16 ________ 2. 12 to 4 ________ 3. 15 : 75 ________

4. [pic] ________ 5. 150 to 15 ________ 6. [pic] ________

Write each ratio in lowest terms.

7. 15 milliliters to 24 liters ________ 8. 6 feet to 15 inches ________

9. 75 cm to 4 m ________ 10. 3 days to 9 hours _________

11. A soccer team played 25 games and won 17.

a. What is the ratio of the number of wins to the number of loses?

b. What is the ratio of the number of games played to the number of games won?

12. In a senior class, there are b boys and g girls. Express the ratio of the number of boys to the

total number in the class.

13. Two numbers are in a ratio of 5 : 3. Their sum is 80. Find the largest number.

14. Mr. Smith and Mr. Kelly are business partners. They agreed to divide the profits in the ratio of 3 : 2. The profit amounted to $24,000. How much did each person receive?

Geometry G Name ________________________

Ratios Worksheet 2 Period ______ Date _____________

Express each ratio in lowest terms.

1. [pic] ________ 2. 96 : 100 ________ 3. 625 to 125 ________

4. 72 to 60 ________ 5. [pic] ________ 6. 49 : 35 ________

7 15 kg to 90 kg ________ 8 18 feet to 4 yards ________

9. 45 meters to 80 meters ________ 10. 10 seconds to 2 minutes ________

11. The Yankees won 125 games, the Red Sox won 97 games, and the Mets won 86 games. What is the ratio of wins of the Yankees to the Red Sox to the Mets?

12. The measure of the angles of a triangle are in a ratio of 2 : 3 : 4. Find the number of degrees in the smallest angle of the triangle.

Do the following pairs form a proportion?

13. [pic] and [pic] 14. [pic] and [pic] 15. [pic] and [pic]

Geometry G Name ________________________

Ratios Worksheet 3 Period ______ Date _____________

Solve each proportion. Circle your final answer.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

Geometry G Name ________________________

Ratios Worksheet 4 Period ______ Date _____________

Applications of Proportions

|1. A recipe for 3 dozen cookies calls for 4 cups of flour. How much flour is |2. A certain medication calls for 250 mg for every 75 lbs of body weight. How |

|needed to make 5 dozen cookies? |many milligrams of medication should a 220-lb person take? |

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|3. A 2-inch wound requires 9 inches of suture thread. How long of a thread |4. An apartment building has 24 identical apartments. It took 42.7 gallons of |

|should a nurse have ready to close a 5-inch wound? |paint to paint 3 apartments. How many gallons of paint are needed to paint 21 |

| |apartments? |

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Do the following ratios form a proportion? Meaning, are they equal?

1. [pic] 2. [pic] 3. [pic] 4. [pic]

Sect 9.2 - Changing the Size of Figures

These figures are similar These are not similar

Similar Figures ~ Two polygons are similar if and only if the____________________ angles are ____________________ and the measures of the _______________________ sides are ___________________________.

The symbol __________ means similar.

(ABC ~ (DEF (“triangle ABC is similar to triangle DEF”)

Corresponding Angles Corresponding Sides

are _______________ have _____________________

( ______ ( ( ______ _______ ( _______

( ______ ( ( ______ _______ ( _______

( ______ ( ( ______ _______ ( _______

Scale Factor -

If the scale factor > 1,

If the scale factor < 1,

Example: Find the dimensions of the figure ...

a) using a scale factor of 2. b) using a scale factor of [pic].

Similar figures are enlargements or reductions of each other. The amount of enlargement or reduction needed to change one figure to the other is called the _________________ . The ratio of the lengths of the corresponding sides of similar figures is the ______________________.

Determine if the polygons are similar. Show work to justify your answer.

1) 2) 3)

Find the values of x and y if (JHI~(MLN.

a) Write proportions for the corresponding sides.

b) Write the proportion c) Write the proportion

to solve for x. to solve for y.

Example: ABCD is similar to WXYZ

The similarity ratio of ABCD to WXYZ is ________.

The scale factor of ABCD to WXYZ is _________.

Label the lengths of the missing sides.

ABCDE is similar to QRSTU

The similarity ratio of ABCDE to QRSTU is __________.

The scale factor of ABCDE to QRSTU is _________.

Find the length of each side.

QU ___________

QR ___________

RS ____________

ST ____________

Perimeter of ABCDE______________

Perimeter of QRSTU______________

ratio of perimeter of ABCDE to perimeter of QRSTU _______________

Geometry Name

Chapter 11.1 Scale Factor Worksheet 1

Scale factor of 3

[pic] [pic]

Scale factor of 2/3

[pic] [pic]

Scale factor of 3/4

[pic] [pic]

Geometry Name

Chapter 11.1 Scale Factor Worksheet 2

Goal: To be able to draw a figure with a given scale factor.

Scale factor of 2

Scale factor of [pic]

Scale factor: 4

Scale factor: [pic]

Scale factor: [pic]

Geometry Name

Chapter 11.1 Similar Figures Worksheet 1

1. Given ABCD ~ WXYZ

a. What angles are congruent?

b. Write the proportions that are equal.

2. Given (XYZ~(RST

a. What angles are congruent?

b. Write the proportions that are equal.

3. Explain why the figures are similar and write the similarity statement.

Geometry Name

Chapter 11.1 Similar Figures Worksheet 2

Determine whether the figures are similar. If yes, what is the scale factor that transforms the figure on the left to the figure on the right? Assume the angles are congruent.

1. Similar ? yes no 2. Similar? yes no

If yes, scale factor (left to right) _____ If yes, scale factor (left to right)____

3. Similar ? yes no 4. Similar? yes no

If yes, scale factor (left to right) _____ If yes, scale factor (left to right)____

5. Similar ? yes no 6. Similar? yes no

If yes, scale factor (left to right) _____ If yes, scale factor (left to right)____

Geometry Name

Chapter 11.2 Similar Triangles Worksheet 3

Goal is to understand notation related to similarity and then apply this notation to find a missing side of similar triangles.

Definition of Similar Polygons: Two polygons are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional.

1. [pic] These corresponding angles are congruent:

_______ [pic] ________

_______ [pic] ________

_______ [pic] ________

These corresponding sides are proportional:

2. [pic]

These corresponding angles are congruent:

[pic] _______ [pic] ________

_______ [pic] ________

_______ [pic] ________

These corresponding sides are proportional:

3. [pic]

Which angles are congruent? What sides are proportional?

4. [pic] What proportions are equal?

Find x Find y

5. [pic] What sides are proportional?

x

Find x:

6. [pic] Find AC and OG.

Geometry Name

Chapter 11.2 Similar Triangles Worksheet 4

Find the missing lengths of the similar triangles.

1. [pic]

Step 1: Write the corresponding sides of [pic] and [pic] as a proportion:

[pic]

Step 2: Fill in the numbers and solve for the missing side.

BC = _____________

FD = _____________

2. [pic]

Step 1: Write the corresponding sides of [pic] and [pic] as a proportion:

[pic]

Step 2: Fill in the numbers and solve for the missing side.

AC = _____________ TG = _____________

3. [pic]

Step 1: Write the corresponding sides of [pic] and [pic] as a proportion:

[pic]

Step 2: Fill in the numbers and solve for the missing side.

PM = ____________ QV = ____________

4. [pic]

BD = _________ EC = _________

Geometry Name

Chapter 11.2 Similar Triangles Worksheet 5

Find the missing lengths. (You may get decimals.)

1. 2.

AC = _________ GE = _________ RS = _________ TR = _________

3. [pic]

YZ = _________ WY = _________

Similarity Names________________________

Geometry G

Round Table ______________________________

Find the missing lengths.

|1. |2. |

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|AC = ___________ |WY = ___________ |

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|EG = ___________ |YZ = ___________ |

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| |x = ___________ |

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|RS = ___________ |NP = ___________ |

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|TR = ___________ |QV = ____________ |

Section 9.3 Notes

Methods of proving Triangles Similar

We will look at ways of proving triangles similar.

Recall what similarity means: 1) Corresponding angles are____________

2) The ratios of the measures of corresponding sides are_______________

Postulate: AA to prove triangles similar

Given two corresponding angles congruent, can you prove the triangles similar by AA?

Therefore, AA is a way to prove triangles similar.

The other two ways to prove triangles similar are:

Theorem:

Theorem:

Don’t forget the ~ when proving similar triangles by the three above methods!

*The 3 ways to prove similar triangles are: ________, ________, and ________.

Examples

Decide if each pair of triangles is similar. If they are, write the correspondence in the first blank and the reason in the second blank. If they are NOT similar, write NS in the second blank.

1) [pic]∆ ABC ~ ∆ _________ by __________

2) ∆ ABC ~ ∆ _________ by __________

3) ∆ YXS ~ ∆ _________ by __________

4) ∆ ABC ~ ∆ _________ by __________

Geometry Name

Chapter 11.2 Justifying Similar Triangles Worksheet 6

Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using SSS, SAS, and AA. Make sure you have work to support your answer.

1.

Yes No [pic] ___________________ ~ [pic] ____________________ by ____________________

2.

Yes No [pic] ____________________ ~ [pic] ____________________ by ___________________

3.

Yes No [pic] ____________________ ~ [pic] ____________________ by ___________________

8.

Yes No [pic] ____________________ ~ [pic] ____________________ by ___________________

9. Ryan is 5 feet tall. His shadow is 9 feet long and the shadow of a building is 36 feet long. How

tall is the building? Draw two similar triangles and then determine the height of the building.

Geometry Name

Chapter 11.2 Justifying Similar Triangles Worksheet 7

Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using SSS(, SAS(, and AA(. Make sure you have work to support your answer.

1.

Yes No

[pic] _________ ~ [pic]_________

by ____________

2. Yes No

[pic] _________ ~ [pic]_________

by ____________

3. Yes No

[pic] _________ ~ [pic]_________

by ____________

4. Yes No

[pic] _________ ~ [pic]_________

by ____________

5. Yes No

[pic] _________ ~ [pic]_________

by ____________

Geometry G Name_________________________

Sec 9.4 Notes

Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then the triangle formed is similar to the original triangle.

If [pic] || [pic], then (ABE ~ (ACD

Let’s see why this is true.

If [pic] || [pic], then the corresponding angles which

are congruent are:

(_____ ( (________ and (_____ ( (________.

By AA, ( _________ ~ ( _________.

Examples

Complete the proportions for the given diagram.

a. [pic] b. [pic]

c. [pic]

We can use these proportions to solve for the missing sides of similar triangles..

1. 2.

Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths.

(Also known as the Side-Splitter Theorem.)

If [pic] || [pic], then [pic].

If you need to find either BE or CD, you still need

to use similar triangles. You CANNOT use the

Side-Splitter Theorem to find these two sides since

they are not “split” sides.

Examples

Write and solve proportions to solve for each variable.

1. 2.

3. 4.

Geometry

Chapter 11.6 Proportional Segments between Parallel Lines

Directions: Find the value each variable in the diagrams.

1.

2.

3.

Practice 9.4

Solve for x for each problem.

1] 2]

3] 4]

5] 6]

7] 8]

Name_________________________________________ Date___________ Hour_______

Sect 9.5 – Triangle Midsegments

Use centimeters or degrees to find the measures of the following…

SR = ________

RN = ________

SN = ________

SP = ________

PR = ________ (S = _________ (N = ________

RI = ________ (RPI = _______ ( PIR = ______

IN = ________ (R = _________

PI = ________

Notice anything???

Fill in the measures of all of the sides and angles of the triangle below. Did the same thing occur as above??

[pic]

THEOREM: The __________________ of a triangle is __________the length of the third side and is ________________ to it.

Examples:

1) In the triangle given, A, B, and C are midpoints of the sides of [pic]. If TU=12. UV=16 and TV=20…

a) Find AB, BC, and AC

b) Name the three pairs of parallel segments

[pic]

2) D is the midpoint of [pic]and E is the midpoint of [pic].

[pic]

a. If AD is 8 and AB is 12 find AC, DC, and DE

AC_________ DC___________ DE___________

b. If [pic], and DE is 17.9 Find [pic] and AB

[pic]__________ AB____________

c. If [pic] and AD is 13 and BC is 27, Find [pic], BE and AC

[pic]_________ BE_________ AC___________

Geometry G Name_________________________

Sec 9.6 Notes Proportional Parts and Parallel Lines

Remember the Side-Splitter Theorem?

Theorem: If a line is parallel to one side of a triangle and intersects the other

two sides, it divides those two sides proportionally.

Given: [pic] || [pic]

Prove: [pic]

What happens if there are more than two parallel lines?

Theorem: If three or more parallel lines intersect two transversals, the parallel lines divide the transversals proportionally.

Given: [pic] || [pic] || [pic]

Conclusion: [pic]

Examples:

1. Complete each proportion.

a. [pic]

b. [pic]

c. [pic]

Write and solve a proportion to find the value of x.

2.

3.

4.

5.

Name________________________________________ Date____________ Hour_______

Sect. 9.7 – Perimeters and Similarity

1) Use the Pythagorean Theorem to find AC and DE.

AC = _____________

DE = ____________

2) Find the following ratios.

[pic] [pic] [pic]

3) Are the triangles similar? YES or NO

If YES, name the similarity correspondence. (_________~(__________ by ________

4) Perimeter of (ABC = ______________ Perimeter of (DFE = _______________

5) Find the ratio of [pic]

6) Compare the ratios of part 2 and part 5. What do you notice??

Let’s try another pair of shapes.

Are the shapes similar?

What is the similarity ratio?

Perimeter of ABCD = _______ Perimeter WXYZ = _______ [pic]______

How does the similarity ratio compare to the ratio of perimeters?

If two triangles are similar, then the measures of the corresponding

perimeters are proportional to the measure to the corresponding sides.

If (HIJ ~ ( LMN, then

[pic]

The perimeter of (GEO is 27 and (GEO ~ (MAT. Use ratios to find the value of each variable.

The ratio found by comparing the measures of corresponding sides of similar triangles is called

the _______________________________ or the ______________________________

Find the scale factor for each pair of similar triangles.

1) 2)

(BAM to (HOT = (CUB to (SOX =

(HOT to (BAM = (SOX to (CUB =

The perimeter of (MDF is 84 feet. If (MDF ~ (KNG and the scale factor of (MDF to (KNG is [pic], find the perimeter of (KNG.

Geometry Name

Chapter 11 Review

Write each ratio in lowest terms.

1. 21 in to 18 in 2. 105 inches : 35 feet

Tell whether each pair of ratios forms a proportion.

3. [pic] and [pic] 4. [pic] and [pic]

Solve for x.

5. [pic] 6. [pic] 7. [pic]

Determine whether the figures are similar. If so, what is the scale factor that transforms the figure on the left to the figure on the right?

8. 9.

Yes No Yes No

Scale Factor _____________ Scale Factor _____________

(left to right) (left to right)

Use the grid provided below to draw a figure that is similar to the given figure, with the indicated scale factor.

10. Scale factor of 2 11. Scale factor of [pic] 12. Scale factor of 3

13. Given [pic], find x and y. Show your work.

.

14. Given [pic], find the s and the length FH. Show your work.

Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using SSS~, SAS~, and AA~. Make sure you have work to support your answer.

15.

Yes No

[pic]__________________ ~[pic]__________________

by ___________

16.

Yes No

[pic]__________________ ~[pic]__________________

by ___________

17.

Yes No

[pic]__________________~[pic]____________

by ___________

Solve for x in each of the diagrams. Show your work.

18. 19.

20.

21. The measure of the angles of a triangle are in a ratio of 2 : 3 : 7. Find the number of degrees

in the largest angle of the triangle.

22. The shadow of a 12-foot tree is 18 feet long at the same time the shadow of a boy is 6 feet l

long. How tall is the boy?

23. A pile of kick boards is 4ft. 4 inches tall and is 6 feet away from a sunbather. At 3:00 a nearby

8-foot lifeguard station casts a 14 foot shadow, will the sunbather have to move out of the shade of the pile at 3:00?

Determine if the figures are similar. If the figures are similar, what is the scale factor that transforms the figure on the left to the figure on the right? (Assume that if a pair of angles appears congruent then they are congruent.)

24. Yes No 25. Yes No

Scale Factor ___________ Scale Factor ___________

26. Yes No 27. Yes No

Scale Factor ___________ Scale Factor ___________

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[pic]

a to b

a : b

Cross-multiplying:

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75

84

96

W

S

H

25

28

32

24

6

X

B

T

12

8

M

W

S

B

A

C

120˚

28˚

R

S

Q

120˚

32˚

8

12

10

x

7

9

18

x

x

15

20

12

12

9

15

5

5

3

7

4

10

10

14

4

3

5

5

5

7

7

16

16

10

10

3

3

2

2

4

6

6

8

4

4

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