proportions and similar figures section 2-8. goals goal to find missing lengths in similar figures....
TRANSCRIPT
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Proportions and Similar Figures
Section 2-8
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Goals
Goal• To find missing lengths in
similar figures.• To use similar figures when
measuring indirectly.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.
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Vocabulary
• Similar Figures• Scale Drawing• Scale• Scale model
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What is Similarity?
Not Similar
Similar
Similar
Not Similar
Similar Triangles
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Similar Figures
Figures that have the same shape but not necessarily the same size are similar figures. But what does “same shape mean”? Are the two heads similar? NOT Sim
ilar
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Similar Figures
Similar figures can be thought of as enlargements or reductions with no irregular distortions.– So two figures are similar if one can be enlarged or
reduced so that it is congruent (means the figures have the same dimensions and shape, symbol ≅) to the original.
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Similar Triangles
• When triangles have the same shape but may be different in size, they are called similar triangles.
• We express similarity using the symbol, ~. (i.e. ΔABC ~ ΔPRS)
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Figures that are similar (~) have the same shape but not necessarily the same size.
Example - Similar Triangles
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Similar figures have exactly the same shape but not necessarily the same size.
Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures.
Similar Figures
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When stating that two figures are similar, use the symbol ~. For the triangles above, you can write ∆ABC ~ ∆DEF. Make sure corresponding vertices are in the same order. It would be incorrect to write ∆ABC ~ ∆EFD.
You can use proportions to find missing lengths in similar figures.
Similar Figures
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Reading Math
• AB means segment AB. AB means the length of AB.
• A means angle A. mA the measure of angle A.
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Example 1
• If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.
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Example 1
• Use the similarity statement.
ΔABC ~ RST
Congruent Angles: A R, B S, C T
Answer:
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Your Turn:• If ΔGHK ~ ΔPQR, determine which of the following
similarity statements is not true.
A. HK ~ QR
B.
C. K ~ R
D. H ~ P
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The two triangles below are similar, determine the length of side x.
x
5.4
5
5.7
Example: Finding the length of a Side of Similar Triangles
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x
5.4
5
5.7
x5.75.45 x5.75.22
x3
Example: Continued
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Find the value of x the diagram. ABCDE ~ FGHJK
14x = 35 Use cross products.
Since x is multiplied by 14, divide both sides by 14 to undo the multiplication.
x = 2.5
The length of FG is 2.5 in.
Example: Finding the length of a Side of Similar Figures
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A
B
C
P
Q
R10
6
c
5
4 d
In the figure, the two triangles are similar. What is the length of c?
45
10 c c540 c8
Your Turn:
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A
B
C
P
Q
R10
6
c
5
4 d
In the figure, the two triangles are similar. What is the length of d?
d
6
5
10 d1030 d3
Your Turn:
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Indirect Measurement
• You can use similar triangles and proportions to find lengths that you cannot directly measure in the real world.
• This is called indirect measurement.• If two objects form right angles with the
ground, you can apply indirect measurement using their shadows.
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Similarity is used to answer real life questions.
• Suppose that you wanted to find the height of this tree.
• Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.
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You can measure the length of the tree’s shadow.
10 feet
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Then, measure the length of your shadow.
10 feet 2 feet
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If you know how tall you are, then you can determine how tall
the tree is.
10 feet 2 feet
6 ft
610 2h
2 60h
30h
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The tree is 30 ft tall. Boy, that’s a tall tree!
10 feet 2 feet
6 ft
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When a 6-ft student casts a 17-ft shadow, a flagpole casts a shadow that is 51 ft long. Find the height of the flagpole.
Set up a proportion for the similar triangles.
17h = 6 • 51 Write the cross products.
h = 18 Simplify.
Divide each side by 17.17h17
6 • 5117=
The height of the flagpole is 18 ft.
Words
Let h = the flagpole’s height.
Proportion
flagpole’s heightstudent’s height
length of flagpole’s shadowlength of student’s shadow=
h6
5117=
Example: Indirect Measurement
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h
6
17102
h
6
102
17
h36
When a 6-ft student casts a 17-ft shadow, a tree casts a shadow that is 102 ft long. Find the height of the tree.
Your Turn:
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1. Use the similar triangles to find the height of the telephone pole.
2. On a sunny afternoon, a goalpost casts a 75 ft
shadow. A 6.5 ft football player next to the goal
post has a shadow 19.5 ft long. How tall is the
goalpost?25 feet
20 feet
8 ft
6 ft
x
15 ft
Your Turn:
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Definition
• Proportions are used to create scale drawings and scale models.
• Scale - a ratio between two sets of measurements, such as 1 in.:5 mi.
• Scale Drawing or Scale Model - uses a scale to represent an object as smaller or larger than the actual object. – A map is an example of a scale drawing.
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A contractor has a blueprint for a house drawn to the scale 1 in.:3 ft.
A wall on the blueprint is 6.5 inches long. How long is the actual wall?
x 1= 3(6.5)x = 19.5
The actual length is 19.5 feet.
Write the scale as a fraction.
Let x be the actual length.
Use cross products to solve.
Example: Scale Drawing
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Write the scale as a fraction.
Let x be the blueprint length.
x 3 = 1(12)
x = 4The blueprint length is 4 inches.
Use cross products to solve.
A contractor has a blueprint for a house drawn to the scale 1 in.:3 ft.
A wall in the house is 12 feet long. How long is the wall on the blueprint?
Example: Scale Drawing
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A scale written without units, such as 32:1, means that 32 units of any measure corresponds to 1 unit of that same measure.
Reading Math
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The actual distance between North Chicago and Waukegan is 4 mi. What is the distance between these two locations on the map?
18x = 4
x ≈ 0.2
The distance on the map is about 0.2 in.
Write the scale as a fraction.
Let x be the map distance.
Use cross products to solve.
Your Turn:
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A scale model of a human heart is 16 ft long. The scale is 32:1 How many inches long is the actual heart that the model represents?
32x = 16
x = 0.5
The actual heart is 0.5 feet or 6 inches.
Write the scale as a fraction.
Let x be the actual distance.
Use cross products to solve.
Your Turn:
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Joke Time• What kind of coffee was served on the Titanic?• Sanka.
• And what kind of lettuce was served on the Titanic?
• Iceberg.
• Why do gorillas have big nostrils?• Because they have big fingers.
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Assignment
2.8 Exercises Pg. 147 – 149: #6 – 38 even