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.
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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 ___________
-----------------------
[pic]
a to b
a : b
Cross-multiplying:
If [pic] , then [pic]
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b. What is the scale factor from right to left?
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A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side and half its length.
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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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