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Proportions with Perimeter, Area, and Volume
Chapter 11.5
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[You will need graph paper]
Objective To discover the relationship between the perimeters, areas and volumes of similar figures [You will need graph paper]
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On Graph Paper Draw rectangle ABCD with length and width of 16 and 12
Draw rectangle EFGH with length and width of 12 and 9 Write a similarity statement for the two rectangles
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Yours should look something like these
F E D C B A 12 16 9 ABCD~EFGH
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Remember, this is called the
Ratios H G F E D C B A 12 16 9 Compare the larger rectangle to the smaller rectangle. Write the ratio of any two corresponding sides. Remember, this is called the LINEAR Ratio AKA: Similarity Ratio Scale Factor
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Ratios H G F E D C B A 12 16 9 P=56 P=42 Calculate the perimeter of each rectangle What is the ratio of the larger perimeter to the smaller?
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Ratios Linear Similarity Conjecture
H G F E D C B A 12 16 9 P=56 P=42 Since the perimeter is a LINEAR measurement, it is in the same LINEAR ratio. Linear Similarity Conjecture The ratios of any corresponding linear measures of similar figures are equal to the ratio of corresponding sides
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Linear Measurements
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Linear ratio MUST be in simplest form
Linear Measurements Perimeter Radius Length Circumference Width Height Diameter Linear ratio MUST be in simplest form Now back to the rectangles…
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Area Calculate the Area of each rectangle What is the ratio of the larger to smaller areas? H G F E D C B A 12 16 9 A=192 A=108
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Area How does this ratio compare to the linear ratio? (The linear ratio was ) H G F E D C B A 12 16 9 A=192 A=108
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Area Draw another set of similar rectangles on your paper and see if your theory works again (try starting with a 5x7 rectangle and choosing a scale factor to make a second rectangle)
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Linear ratio MUST be in simplest form
Area Proportional Areas Conjecture If corresponding sides of two similar polygons or the radii of two circles compare in the ratio , then their areas compare in the ratio Linear ratio MUST be in simplest form
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Check This Out (?) NCTM Applet for Perimeter and Area
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Area Examples Linear (L): Area (A):
The ratio of the corresponding midsegments of two similar trapezoids is 4:5. What is the ratio of their areas? Linear (L): Area (A):
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Find the Linear and Area ratios
Area Examples Find the Linear and Area ratios L: A:
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Area Examples Fun Fact: ALL circles are similar! L: A: A=560π cm
What is the area of circle N (in terms of π)? L: A: q m P N A=560π cm
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Volume Consider these rectangular prisms Are all of their corresponding linear measures proportional? What is the linear ratio? What is the area ratio? L: A: FIX ME!!! 1.5 1 4.5 3 2 3
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Volume Find the volume of each prism What is the ratio of the volumes? 1.5 V=20.25 1 V=6 4.5 3 2 3
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Volume L: A: and the ratio of VOLUMES is Is there a relationship? 1.5 V=20.25 1 V=6 4.5 3 2 3
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Volume Proportional Volumes Conjecture If corresponding edges (or radii, height, etc.) of two similar solids compare in the ratio , then their areas compare in the ratio
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Check This Out (?) NCTM Applet for Volume of Similar Solids
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Volume Examples L: A: V:
The corresponding heights of two similar cylinders is 2:5. What are the Linear, Area and Volume ratios? L: A: V: FIND ALL ratios!! 5 2
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Volume Examples X Y L: A: VX = 35.1ft3 V: 9ft k
Triangular prisms X and Y are similar The linear ratio is ¾. Find the area and volume ratios. L: A: VX = 35.1ft3 X Y V: 9ft k
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Volume Examples X Y L: A: V: VX = 35.1ft3 9ft k
Triangular prisms X and Y are similar The linear ratio is ¾. Find K L: A: V: VX = 35.1ft3 X Y 9ft k
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Volume Examples X Y L: A: V: VX = 35.1ft3 9ft k
Triangular prisms X and Y are similar The linear ratio is ¾. Find volume of prism Y L: A: V: VX = 35.1ft3 X Y 9ft k
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You can’t jump between area and volume without going through linear
The Ratios Linear Area Volume You can’t jump between area and volume without going through linear
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