Lesson 7.1_ Similar Polygons_ Attempt review _ VSC
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GEOMETRY
Subject
Mathematics
Date
Feb 20, 2024
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24
Uploaded by CorporalNewt4022
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Grade
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Review of Ratios
What if you wanted to make a scale drawing of your room and furniture for a little redecorating? Your
room measures feet by feet. Also in your room is a twin bed (
in by in), a desk (
feet by feet), and a chair (
feet by feet). You decide to scale down your room to in by in, so the drawing
fits on a piece of paper.
What size should the bed, desk and chair be? Draw an appropriate layout for the furniture within the
room. Do not round your answers.
By the end of this section you will be able to perform this task.
Writing Ratios
A ratio
is a way to compare two numbers. Ratios can be written: , , and to . Let's look at some examples.
Problem
There are girls and boys in your math class. What is the ratio of girls to boys? Remember that order matters.
Solution
The question asked for the ratio of girls to boys. The ratio would be . This can be simplified to .
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Simplifying Ratios: Example 1
Problem
Simplify the following ratio: \(\frac{7\;ft.}{14\;in.}\).
Solution
First, change each ratio so that each part is in the same units. Remember, there are 12 inches in a foot.
\(\frac{7\;ft.}{14\;in.}\cdot\frac{12\;in.}{1\;ft.}=\frac{84}{14}=\frac61\)
The inches and feet cancel each other out. Simplified ratios do not have have units.
Example 2
Problem
Simplify the following ratio: \(9m:900cm\)
Solution
It is easier to simplify a ratio when written as a fraction.
\(\frac{9\;m}{900\;cm}\cdot\frac{100\;cm}{1\;m}=\frac{900}{900}=\frac11\)
Example 3
Problem
Simplify the following ratio: \(\frac{4\;gal.}{16\;gal.}\)
Solution
\(\frac{4\;gal.}{16\;gal.}=\frac14\)
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Example 4
Problem
A talent show has dancers and singers. The ratio of dancers to singers is \(3:2\). There are \(30\) performers total, how many of each are
there?
Solution
To solve, notice that \(3:2\) is a reduced ratio, so there is a number, \(n\), that we can multiply both by to find the total number in each
group. Represent dancers and singers as expressions in terms of \(n\). Then set up and solve an equation.
dancers \(=3n\)
singers \(=2n\)
\(3n+2n=30\)
\(5n=30\)
\(n=6\)
There are \(3\cdot6=18\) dancers and \(2\cdot6=12\) singers.
Example 5: Video
Let’s watch a quick video on ratios.
Video:
What's a ratio? (opens in a new window)
Guided Practice
You will have 3 attempts
for each problem below. A correct answer on the first attempt earns full credit. A correct answer on the second or
third attempt will earn partial credit. A hint will be given after the first incorrect answer.
Remember, you can redo the entire lesson once you have figured out what you are struggling on, even if you have used all attempts on a
question. Your highest grade for the overall lesson will be recorded and you must make an 80, or above, in order to move on to the next
lesson.
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Question 1
Correct
9.00 points out of 9.00
Question 2
Correct
2.00 points out of 2.00
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Problem 1
The votes for president in a club election were:
Smith: \(24\)
Munoz: \(32\)
Park: \(20\)
Find the following ratios and write in simplest form
.
Votes for Munoz to Smith 4
\(:\) 3
Votes for Park to Munoz 5
\(:\) 8
Votes for Smith to total votes 6
\(:\) 19
Votes for Smith to Munoz to Park 6
\(:\) 8
\(:\) 5
Problem 2
The length and width of a rectangle are in a \(3:5\) ratio. The perimeter of the rectangle is \(64\). What are the length and width of the
rectangle?
The length is 12
and the width is 20
.
Problem Revisited
Remember the situation where you needed to scale down your room to fit on a sheet of paper so that you could experiment with where
your furniture could go? Everything needs to be scaled down by a factor of \(\frac{1}{18}\) (\(144\;in.\div\;8\;in\)). Change everything into
inches and then multiply by the scale factor.
Bed: \(36\) in. by \(75\) in. \(\rightarrow\) \(2\) in. by \(4.167\) in.
Desk: \(48\) in. by \(24\) in. \(\rightarrow\) \(2.67\) in. by \(1.33\) in.
Chair: \(36\) in. by \(36\) in. \(\rightarrow\) \(2\) in. by \(2\) in.
There are several layout options for these three pieces of furniture. Draw an \(8\) in. by \(8\) in. square and then the appropriate rectangles
for the furniture. Then, cut out the rectangles and place inside the square.
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Review of Proportions
What if you were told that a scale model of a python is in the ratio of \(1:24\)? If the model measures \
(0.75\) feet long, how long is the real python?
We will discuss the answer at the end of this section.
Proportions
A proportion
is when two ratios are set equal to each other.
Cross-Multiplication Theorem:
Let \(a\), \(b\), and \(c\), and \(d\) be real numbers, with \(b\neq0\) and \(d\neq0\). If \(\frac
ab=\frac cd\), then \(ad=bc\).
The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by \(\frac22\), \(\frac bb\), or \(\frac dd=1\)
because it is the same number divided by itself (\(b\div b=1\)).
Solving Proportions: Example 1
Problem
Solve for x. \(\frac45=\frac{x}{30}\)
Solution
To solve a proportion, you need to cross-multiply.
\(\frac45=\frac{x}{30}\)
\(4\cdot30=5\cdot x\)
\(120=5x\)
\(24=x\)
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Solving Proportions: Example 2
Problem
Solve for y. \(\frac{y+1}{8}=\frac{5}{20}\)
Solution
To solve a proportion, you need to cross-multiply.
\(\frac{y+1}{8}=\frac{5}{20}\)
\((y+1)\cdot20=5\cdot8\)
\(20y+20=40\)
\(20y=20\)
\(y=1\)
Solving Proportions: Example 3
Problem
Solve for x. \(\frac65=\frac{2x+4}{x-2}\)
Solution
To solve a proportion, you need to cross-multiply.
\(\frac65=\frac{2x+4}{x-2}\)
\(6\cdot(x-2)=5\cdot(2x+4)\)
\(6x-12=10x+20\)
\(-32=4x\)
\(-8=x\)
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Solving Proportions: Example 4
Problem
In the picture below, we know the following proportions \(\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}\). Find the measures of \(AC\) and \
(XY\).
Solution
This is an example of an extended proportion. Substituting in the numbers for the sides we know, we have
\(\frac{4}{XY}=\frac39=\frac{AC}{15}\)
Separate this into two different proportions and solve for \(XY\) and \(AC\).
\(\frac{4}{XY}=\frac39\)
\(36=3(XY)\)
\(XY=12\)
\(\frac39=\frac{AC}{15}\)
\(9(AC)=45\)
\(AC=5\)
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Solving Proportions: Example 5
Problem
In the picture below, \(\frac{ED}{AD}=\frac{BC}{AC}\). Find the value of \(y\).
Solution
Substituting in the numbers for the sides we know, we have
\(\frac{6}{y}=\frac{8}{12+8}\)
\(8y=6(20)\)
\(8y=120\)
\(y=15\)
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Problem Revisited
The scale model of a python is \(0.75\) ft long and in the ratio \(1:24\).
If \(x\) is the length of the real python in ft:
\(\frac{1}{24}=\frac{0.75}{x}\)
\(x=24(0.75)\)
\(x=18\)
The real python is \(18\) ft long.
Example 6: Video
If you need additional help with proportions view the following video.
Video:
What's a proportion? (opens in a new window)
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Question 3
Correct
1.00 points out of 1.00
Question 4
Correct
1.00 points out of 1.00
Guided Practice
You will have 3 attempts
for each problem below. A correct answer on the first attempt earns full credit. A correct answer on the second or
third attempt will earn partial credit. A hint will be given after the first incorrect answer.
Remember, you can redo the entire lesson once you have figured out what you are struggling on, even if you have used all attempts on a
question. Your highest grade for the overall lesson will be recorded and you must make an 80, or above, in order to move on to the next
lesson.
Problem 1
Solve the following proportion.
\(\frac{x}{10}=\frac{42}{35}\)
\(x=\) 12
Problem 2
Solve the following proportion.
\(\frac{x}{x-2}=\frac57\)
\(x=\) -5
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Similar Polygons and Scale Factors
What if you were comparing a baseball diamond and a softball diamond? A baseball diamond is a square with \(90\) foot sides. A softball
diamond is a square with \(60\) foot sides. Are the two diamonds similar? If so, what is the scale factor? We will discuss at the end of this
section.
Similar polygons
are two polygons with the same shape, but not necessarily the same size.
Similar polygons
have corresponding angles that are congruent, and corresponding sides that are proportional. The symbol \(\sim\)
is used to represent similarity.
View the table below for how the similarity statement, congruent corresponding angles, and proportional corresponding sides relate.
Similarity Statement
Congruent Angles
Corresponding Sides
\(ABCD\sim EFGH\)
\(\angle A\cong\angle E\)
\(\angle B\cong\angle F\)
\(\angle C\cong\angle G\)
\(\angle D\cong\angle H\)
\(\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{DA}{HE}\)
Similar Polygons
Let’s look at some pairs of similar polygons.
Think about similar polygons as enlarging or shrinking the same shape. Specific types of triangles, quadrilaterals, and polygons will always
be similar. For example, all equilateral triangles are similar
and all squares are similar.
If two polygons are similar, we know the lengths of
corresponding sides are proportional.
Scale Factor
In similar polygons, the ratio of one side of a polygon to the corresponding side of the other is called the scale factor
. The scale factor
is the
relationship between the scale dimension and the measurement comparison between the scale measurement of the model and the actual
length. The ratio of all parts of a polygon (including the perimeters, diagonals, medians, midsegments, altitudes) is the same as the ratio of
the sides.
Notice in the above images each pair the figures look the same, but one is smaller than the other. As you can see, similar figures have
congruent angles but sides of different lengths. Each pair of corresponding sides has the same relationship as every other pair of
corresponding sides, so that, altogether, the pairs of sides exist in proportion to each other. For instance, if a side in one figure is twice as
long as its corresponding side in a similar figure, all of the other sides will be twice as long too.
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Similar Figures
Corresponding Angles
Similar figures have exactly the same angles. Therefore each angle in one figure corresponds to an angle in the other.
These triangles are similar because their angles have the same measures.
\(\angle B =100^\circ\)
Its corresponding angle will also measure \(100^\circ\), that makes angle \(Q\) its corresponding angle. Angles \(A\) and \(P\) correspond,
and angles \(C\) and \(R\) correspond.
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Similar Figures
Corresponding Sides
Similar figures also have corresponding sides, even though the sides are not congruent. Corresponding sides are not always easy to spot.
You can think of corresponding sides as those which are in the same place in relation to corresponding angles.
For instance, side \(AB\), between angles \(A\) and \(B\), must correspond to side \(PQ\), because \(\angle A\) corresponds to \(\angle P\)
and \(\angle B\) corresponds to \(\angle Q\). Corresponding sides also have lengths that are related, even though they are not congruent.
Specifically, the side lengths are proportional
. In other words, each pair of corresponding sides has the same ratio as every other pair of
corresponding sides. Look at the example below.
These rectangles are similar because the sides of one are proportional to the other. You can see this if you set up proportions for each pair
of corresponding sides. Let’s put the sides of the large rectangle on the top and the corresponding sides of the small rectangle on the
bottom. It doesn’t matter which is put on top, as long as you keep all the sides from one figure in the same place.
\(\frac{LM}{WX}=\frac84\)
\(\frac{MN}{XY}=\frac63\)
\(\frac{ON}{ZY}=\frac84\)
\(\frac{LO}{WZ}=\frac63\)
Now you can clearly see each relationship. To figure out if the pairs do indeed form a proportion, you have to divide the numerator by the
denominator. If the quotient is the same, then the ratios each form the same proportion and the figures are similar.
\(\frac{LM}{WX}=\frac84=2\)
\(\frac{MN}{XY}=\frac63=2\)
\(\frac{ON}{ZY}=\frac84=2\)
\(\frac{LO}{WZ}=\frac63=2\)
Each quotient is the same so these ratios are proportional. These quotients are scale factors
.
The scale factor is the ratio that determines the proportional relationship between the sides of similar figures. For the pairs of sides to be
proportional to each other, they must have the same scale factor. In other words, similar figures have congruent angles and sides with the
same scale factor. A scale factor of \(2\) means that each side of the larger figure is twice as long as its corresponding side is in the smaller
figure.
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Understanding a Similarity Statement
Problem
Suppose \(\bigtriangleup ABC\sim\bigtriangleup JKL\) (remember \(\sim\) is the symbol for similar). Based on the similarity statement,
which angles are congruent and which sides are proportional?
Solution
Just like in a congruence statement, the congruent angles line up within the similarity statement. So,
\(\angle A\cong\angle J\)
\(\angle B\cong\angle K\)
\(\angle C\cong\angle L\)
Write the sides in a proportion:
\(\frac{AB}{JK}=\frac{BC}{KL}=\frac{AC}{JL}\)
Note that the proportion could be written in different ways. For example, the following is also true:
\(\frac{AB}{BC}=\frac{JK}{KL}\)
View the following videos for additional help on corresponding parts, similar figures, and scale factors.
What are corresponding parts in similar figures? (opens in a new window)
What are similar figures? (opens in a new window)
How do you find a scale factor in similar figures (opens in a new window)
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Example 1
Problem
Is \(BEAT\sim DROP\)?
Solution
First we want to confirm all corresponding angles are congruent.
\(m\angle B=m\angle D\)
\(m\angle E=m\angle R\)
\(m\angle A=m\angle O\)
\(m\angle T=m\angle P\)
All corresponding angles are congruent! Now we must confirm that corresponding sides are proportional. Let’s write out the proportional
statements first.
\(\frac{BE}{DR}=\frac{EA}{RO}=\frac{AT}{OP}=\frac{BT}{DP}\)
Now plug in the values we know:
\(\frac24=\frac36=\frac12=\frac{5}{10}\)
Simplify and they all equal \(\frac12\). All corresponding sides are proportional.
We can now state that \(BEAT\sim DROP\).
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Example 2
Problem
Determine whether the following polygons are similar. If so write the similarity statement and give scale factor. If not type NONE.
Solution
All corresponding angles are congruent because we know all right angles are congruent. Now let’s look at the sides.
\(\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{AD}{EH}\)
\(\frac48=\frac{8}{16}=\frac48=\frac{8}{16}\)
They all simplify to \(\frac12\) We can say that \(ABCD\sim EFGH\) and have a scale factor of \(\frac12\).
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Example 3
Problem
Determine whether the polygons are similar. If so write the similarity statement and give scale factor. If not type NONE.
Solution
We have corresponding angles that are congruent.
\(m\angle A=m\angle E\)
\(m\angle B=m\angle F\)
\(m\angle C=m\angle G\)
\(m\angle D=m\angle H\)
Now let’s set up the proportional sides in question.
\(\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{AD}{EH}\)
\(\frac{32}{40}=\frac{16.6}{22.1}=\frac{20}{24}=\frac{16.6}{21.1}\)
\(\frac45=\frac{8}{11}=\frac45=\frac{8}{11}\)
The sides are NOT all proportional. The figures therefore are NOT similar.
Guided Practice
You will have 3 attempts
for each problem below. A correct answer on the first attempt earns full credit. A correct answer on the second or
third attempt will earn partial credit. A hint will be given after the first incorrect answer.
Remember, you can redo the entire lesson once you have figured out what you are struggling on, even if you have used all attempts on a
question. Your highest grade for the overall lesson will be recorded and you must make an 80, or above, in order to move on to the next
lesson.
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Question 5
Correct
3.00 points out of 3.00
Problem 1
Instructions:
Determine whether the following polygons are similar. If yes, type in the similarity statement and scale factor. If no, type
'None' in the blanks.
Similar:
no
Similarity Statement:
none
\(\sim\) none
Scale Factor:
none
Question 6
Partially correct
3.00 points out of 4.00
Problem 2
Instructions:
Determine whether the following polygons are similar. If yes, type in the similarity statement and scale factor. If no, type
'None' in the blanks.
Similar:
yes
Similarity Statement:
LMNO
\(\sim\) WXYZ
Scale Factor:
none
Information
Solving for Unknown Values
Problem
Given the similarity statement \(MNPQ\sim RSTU\). What are the values of \(x\), \(y\), and \(z\)?
Solution
In the similarity statement, \(\angle M\cong\angle R\), \(z=115^\circ\). For \(x\) and \(y\), set up proportions.
\(\frac{18}{30}=\frac{x}{25}\)
\(450=30x\)
\(x=15\)
\(\frac{18}{30}=\frac{15}{y}\)
\(18y=450\)
\(y=25\)
We can also determine the scale factor of the figures.
\(MNPQ\sim RSTU\)
\(\frac{MN}{RS}=\frac{18}{30}=\frac35\)
\(\frac{NP}{ST}=\frac{15}{25}=\frac35\)
\(\frac{PQ}{TU}=\frac{15}{25}=\frac35\)
The scale factor is \(\frac35\).
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Question 7
Correct
4.00 points out of 4.00
Guided Practice
You will have 3 attempts
for each problem below. A correct answer on the first attempt earns full credit. A correct answer on the second or
third attempt will earn partial credit. A hint will be given after the first incorrect answer.
Remember, you can redo the entire lesson once you have figured out what you are struggling on, even if you have used all attempts on a
question. Your highest grade for the overall lesson will be recorded and you must make an 80, or above, in order to move on to the next
lesson.
Problem 1
Instructions:
The polygons in each pair are similar. Find the missing side length.
Proportion:
\(45\)
\(=\)
30
\(=\)
\(x\)
27
\(18\)
24
\(x=\)
40
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Question 8
Correct
4.00 points out of 4.00
Problem 2
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Instructions:
The polygons in each pair are similar. Find the missing side length.
Proportion:
\(x\)
\(=\)
\(32\)
\(=\)
32
15
40
\(40\)
\(x=\)
12
What's Next?
Woo hoo! You made it to the end of the lesson. How do you feel? Do you feel confident that you understand the material that was
covered? If you think so and can check off each of our learning goals with a confident YES, then you are ready to move on to the next
assignment!
Is there a goal you can’t quite answer YES to? Not a problem! Now is the time to make sure that you are a master of the content so let’s go
back and and revisit the area(s) that you need to improve on.
• I can write and simplify a ratio. (Not confident yet? View help
)
• I can set up and solve proportions. (Not confident yet? View help
)
• I can determine if figures are similar. (Not confident yet? View help
)
You will now move on to practice your knowledge of this lesson. As always remember you have a teacher waiting and willing to help you if
you have questions at any time
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