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Eureka Math⢠Homework Helper
2015â2016
Grade 8 Module 3
Lessons 1â14
2015-16
Lesson 1: What Lies Behind âSame Shapeâ?
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G8-M3-Lesson 1: What Lies Behind âSame Shapeâ?
1. Let there be a dilation from center ðð. Then, ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð) = ððâ², and ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð) = ððâ². Examine thedrawing below. What can you determine about the scale factor of the dilation?
The dilation must have a scale factor larger than ðð, ðð > ðð, since the dilated points are farther from the center than the original points.
2. Let there be a dilation from center ðð with a scale factor ðð = 2. Then, ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð) = ððâ², andð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð) = ððâ². |ðððð| = 1.7 cm, and |ðððð| = 3.4 cm, as shown. Use the drawing below to answerparts (a) and (b). The drawing is not to scale.
a. Use the definition of dilation to determine |ððððâ²|.|ð¶ð¶ð¶ð¶â²| = ðð|ð¶ð¶ð¶ð¶|; therefore, |ð¶ð¶ð¶ð¶â²| = ðð â (ðð.ðð) = ðð.ðð and |ð¶ð¶ð¶ð¶â²| = ðð.ðð ðððð.
b. Use the definition of dilation to determine |ððððâ²|.|ð¶ð¶ð¶ð¶â²| = ðð|ð¶ð¶ð¶ð¶|; therefore, |ð¶ð¶ð¶ð¶â²| = ðð â (ðð.ðð) = ðð.ðð and |ð¶ð¶ð¶ð¶â²| = ðð.ðð ðððð.
I remember from the last module that the original points are labeled without primes, and the images are labeled with primes.
I know that the bars around the segment represent length. So, |ððððâ²| is said, âThe length of segment ðððð prime.â
We talked about the definition of dilation in class today. I should check the Lesson Summary box to review the definition.
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Lesson 1: What Lies Behind âSame Shapeâ?
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3. Let there be a dilation from center ðð with a scale factor ðð. Then, ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðµðµ) = ðµðµâ², ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ð¶ð¶) = ð¶ð¶â², and |ðððµðµ| = 10.8 cm, |ððð¶ð¶| = 5 cm, and |ðððµðµâ²| = 2.7 cm, as shown. Use the drawing below to answer parts (a)â(c).
a. Using the definition of dilation with |ðððµðµ| and |ðððµðµâ²|, determine the scale factor of the dilation.
|ð¶ð¶ð¶ð¶â²| = ðð|ð¶ð¶ð¶ð¶|, which means ðð.ðð = ðð â (ðððð.ðð), then
ðð.ðððððð.ðð
= ðð ðððð
= ðð
b. Use the definition of dilation to determine |ððð¶ð¶â²|.
Since the scale factor, ðð, is ðððð, then |ð¶ð¶ð¶ð¶â²| = ðð
ðð|ð¶ð¶ð¶ð¶|;
therefore, |ð¶ð¶ð¶ð¶â²| = ðð
ððâ ðð = ðð.ðððð, and |ð¶ð¶ð¶ð¶â²| =
ðð.ðððð ðððð.
Since |ðððµðµâ²| = ðð|ðððµðµ|, then by the multiplication property of
equality, ï¿œðððµðµâ²ï¿œ
|ðððµðµ| = ðð.
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Lesson 2: Properties of Dilations
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G8-M3-Lesson 2: Properties of Dilations
1. Given center ðð and quadrilateral ðŽðŽðŽðŽðŽðŽðŽðŽ, use a ruler to dilate the figure from center ðð by a scale factor of ðð = 1
4. Label the dilated quadrilateral ðŽðŽâ²ðŽðŽâ²ðŽðŽâ²ðŽðŽâ².
The figure in red below shows the dilated image of ðšðšðšðšðšðšðšðš. All measurements are in centimeters.
|ð¶ð¶ðšðšâ²| = ðð|ð¶ð¶ðšðš| |ð¶ð¶ðšðšâ²| = ðð|ð¶ð¶ðšðš| |ð¶ð¶ðšðšâ²| = ðð|ð¶ð¶ðšðš| |ð¶ð¶ðšðšâ²| = ðð|ð¶ð¶ðšðš|
|ð¶ð¶ðšðšâ²| = ðððð
(ðððð.ðð) |ð¶ð¶ðšðšâ²| = ðððð
(ðð.ðð) |ð¶ð¶ðšðšâ²| = ðððð
(ðð.ðð) |ð¶ð¶ðšðšâ²| = ðððð
(ðð.ðð)
|ð¶ð¶ðšðšâ²| = ðð.ðð |ð¶ð¶ðšðšâ²| = ðð.ðð |ð¶ð¶ðšðšâ²| = ðð.ðð |ð¶ð¶ðšðšâ²| = ðð.ðð
I need to draw the rays from the center ðð to each point on the figure and then measure the length from ðð to each point.
Now that I have computed the lengths from the center to each of the dilated points, I can find the vertices of the dilated figure.
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Lesson 2: Properties of Dilations
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2. Use a compass to dilate the figure ðŽðŽðŽðŽðŽðŽ from center ðð, with scale factor ðð = 3.
The figure in red below shows the dilated image of ðšðšðšðšðšðš.
I need to draw the rays like before, but this time I can use a compass to measure the distance from the center ðð to a point and again use the compass to find the length 3 times from the center.
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Lesson 2: Properties of Dilations
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3. Use a compass to dilate the figure ðŽðŽðŽðŽðŽðŽðŽðŽ from center ðð, with scale factor ðð = 2. a. Dilate the same figure, ðŽðŽðŽðŽðŽðŽðŽðŽ, from a new center, ððâ², with scale factor ðð = 2. Use double primes
(ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â²) to distinguish this image from the original.
The figure in blue, ðšðšâ²ðšðšâ²ðšðšâ²ðšðšâ², shows the dilation of ðšðšðšðšðšðšðšðš from center ð¶ð¶, with scale factor ðð = ðð. The figure in red, ðšðšâ²â²ðšðšâ²â²ðšðšâ²â²ðšðšâ²â², shows the dilation of ðšðšðšðšðšðšðšðš from center ð¶ð¶â² with scale factor ðð = ðð.
Since the original image was dilated by a scale factor of 2 each time, I know that the dilated figures are congruent.
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Lesson 2: Properties of Dilations
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b. What rigid motion, or sequence of rigid motions, would map ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â² to ðŽðŽâ²ðŽðŽâ²ðŽðŽâ²ðŽðŽâ²?
A translation along vector ðšðšâ²â²ðšðšâ²ï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œâ (or any vector that connects a point of ðšðšâ²â²ðšðšâ²â²ðšðšâ²â²ðšðšâ²â² and its corresponding point of ðšðšâ²ðšðšâ²ðšðšâ²ðšðšâ²) would map the figure ðšðšâ²â²ðšðšâ²â²ðšðšâ²â²ðšðšâ²â² to ðšðšâ²ðšðšâ²ðšðšâ²ðšðšâ².
The image below (with rays removed for clarity) shows the vector ðšðšâ²â²ðšðšâ²ï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œâ .
4. A line segment ðŽðŽðŽðŽ undergoes a dilation. Based on todayâs lesson, what is the image of the segment?
The segment dilates as a segment.
5. â ðŽðŽðððŽðŽ measures 24°. After a dilation, what is the measure of â ðŽðŽâ²ðððŽðŽâ²? How do you know?
The measure of â ðšðšâ²ð¶ð¶ðšðšâ² is ðððð°. Dilations preserve angle measure, so â ðšðšâ²ð¶ð¶ðšðšâ² remains the same measure as â ðšðšð¶ð¶ðšðš.
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Lesson 3: Examples of Dilations
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G8-M3-Lesson 3: Examples of Dilations
1. Dilate the figure from center ðð by a scale factor of ðð = 3. Make sure to use enough points to make a good image of the original figure.
The dilated image is shown in red below. Several points are needed along the curved portion of the diagram to produce an image similar to the original.
If I only dilate the points ðµðµ, ðžðž, and ð·ð·, then the dilated figure will look like a triangle. I will need to dilate more points along the curve ðµðµð·ð·.
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Lesson 3: Examples of Dilations
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2. A triangle ðŽðŽðµðµðŽðŽ was dilated from center ðð by a scale factor of ðð = 8. What scale factor would shrink the dilated figure back to the original size?
A scale factor of ðð = ðð
ðð would bring the dilated figure
back to its original size.
3. A figure has been dilated from center ðð by a scale factor of ðð = 23. What scale factor would shrink the
dilated figure back to the original size?
A scale factor of ðð = ðð
ðð would bring the dilated figure back to its original size.
To dilate back to its original size, I need to use the reciprocal of the original scale factor because ðð â 1
ðð= 1.
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Lesson 4: Fundamental Theorem of Similarity (FTS)
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G8-M3-Lesson 4: Fundamental Theorem of Similarity (FTS)
1. In the diagram below, points ðð, ðð, and ðŽðŽ have been dilated from center ðð by a scale factor of ðð = 1
5 on a piece of lined paper.
a. What is the relationship between segments ðððð and ððâ²ððâ²? How do you know?
ð·ð·ð·ð·ï¿œï¿œï¿œï¿œ and ð·ð·â²ð·ð·â²ï¿œï¿œï¿œï¿œï¿œï¿œ are parallel because they follow the lines on the lined paper.
b. What is the relationship between segments ðððŽðŽ and ððâ²ðŽðŽâ²? How do you know?
ð·ð·ð·ð·ï¿œï¿œï¿œï¿œ and ð·ð·â²ð·ð·â²ï¿œï¿œï¿œï¿œï¿œï¿œ are also parallel because they follow the lines on the lined paper.
c. Identify two angles whose measures are equal. How do you know?
â ð¶ð¶ð·ð·ð·ð· and â ð¶ð¶ð·ð·â²ð·ð·â² are equal in measure because they are corresponding angles created by parallel lines ð·ð·ð·ð· and ð·ð·â²ð·ð·â².
d. What is the relationship between the lengths of segments ðŽðŽðð and ðŽðŽâ²ððâ²? How do you know?
The length of segment ð·ð·â²ð·ð·â² will be ðððð the length of segment ð·ð·ð·ð·. The FTS states that the length of
the dilated segment is equal to the scale factor multiplied by the original segment length, or |ð·ð·â²ð·ð·â²| = ðð|ð·ð·ð·ð·|.
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Lesson 4: Fundamental Theorem of Similarity (FTS)
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2. Reynaldo sketched the following diagram on graph paper. He dilated points ðµðµ and ð¶ð¶ from center ðð.
a. What is the scale factor ðð? Show your work.
|ð¶ð¶ð©ð©â²| = ðð|ð¶ð¶ð©ð©| ðð = ðð(ðð) ðððð
= ðð
ðð = ðð
b. Verify the scale factor with a different set of segments.
|ð©ð©â²ðªðªâ²| = ðð|ð©ð©ðªðª| ðð = ðð(ðð) ðððð
= ðð
ðð = ðð
c. Which segments are parallel? How do you know?
Segments ð©ð©ðªðª and ð©ð©â²ðªðªâ² are parallel. They are on the lines of the grid paper, which I know are parallel.
d. Which angles are equal in measure? How do you know?
|â ð¶ð¶ð©ð©â²ðªðªâ²| = |â ð¶ð¶ð©ð©ðªðª|, and |â ð¶ð¶ðªðªâ²ð©ð©â²| = |â ð¶ð¶ðªðªð©ð©| because they are corresponding angles of parallel lines cut by a transversal.
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Lesson 4: Fundamental Theorem of Similarity (FTS)
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3. Points ðµðµ and ð¶ð¶ were dilated from center ðð.
a. What is the scale factor ðð? Show your work.
|ð¶ð¶ð©ð©â²| = ðð|ð¶ð¶ð©ð©| ðð = ðð(ðð) ðð = ðð
b. If |ððð¶ð¶| â 2.2, what is |ððð¶ð¶â²|?
|ð¶ð¶ðªðªâ²| = ðð|ð¶ð¶ðªðª|
|ð¶ð¶ðªðªâ²| â ðð(ðð.ðð)
|ð¶ð¶ðªðªâ²| â ðððð
c. How does the perimeter of â³ ðððµðµð¶ð¶ compare to the perimeter of â³ ðððµðµâ²ð¶ð¶â²?
Perimeter â³ð¶ð¶ð©ð©ðªðª â ðð + ðð + ðð.ðð Perimeter â³ ð¶ð¶ð©ð©â²ðªðªâ² â ðð + ðððð + ðððð
Perimeter â³ð¶ð¶ð©ð©ðªðª â ðð.ðð Perimeter â³ ð¶ð¶ð©ð©â²ðªðªâ² â ðððð
d. Was the perimeter of â³ ðððµðµâ²ð¶ð¶â² equal to the perimeter of â³ ðððµðµð¶ð¶ multiplied by scale factor ðð? Explain.
Yes. The perimeter of â³ ð¶ð¶ð©ð©â²ðªðªâ² was five times the perimeter of â³ ð¶ð¶ð©ð©ðªðª, which makes sense because the dilation increased the length of each segment by a scale factor of ðð. That means that each side of â³ð¶ð¶ð©ð©â²ðªðªâ² was five times as long as each side of â³ð¶ð¶ð©ð©ðªðª.
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Lesson 5: First Consequences of FTS
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When we did this in class, I know we started by marking a point ðµðµ on the ð¥ð¥-axis on the same vertical line as point ðŽðŽ. Then we used what we learned in the last lesson. I should review my classwork.
Now I know which vertical line ðŽðŽâ² will fall on; it is the same as where ðµðµâ² is.
G8-M3-Lesson 5: First Consequences of FTS
1. Dilate point ðŽðŽ, located at (2, 4) from center ðð, by a scale factor of ðð = 72. What is the precise location of
point ðŽðŽâ²?
|ð¶ð¶ð©ð©â²| = ðð|ð¶ð¶ð©ð©|
|ð¶ð¶ð©ð©â²| =ðððð
(ðð)
|ð¶ð¶ð©ð©â²| = ðð
|ðšðšâ²ð©ð©â²| = ðð|ðšðšð©ð©|
|ðšðšâ²ð©ð©â²| =ðððð
(ðð)
|ðšðšâ²ð©ð©â²| =ðððððð
|ðšðšâ²ð©ð©â²| = ðððð
I know that segment ðŽðŽâ²ðµðµâ² is a dilation of segment ðŽðŽðµðµ by the same scale factor ðð = 7
2. Since ðŽðŽðµðµï¿œï¿œï¿œï¿œ is
contained within a vertical line, I can easily determine its length is 4 units.
Therefore, ðšðšâ² is located at (ðð,ðððð).
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Lesson 5: First Consequences of FTS
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2. Dilate point ðŽðŽ, located at (7, 5) from center ðð, by a scale factor of ðð = 37. Then, dilate point ðµðµ, located at
(7, 3) from center ðð, by a scale factor of ðð = 37. What are the coordinates of ðŽðŽâ² and ðµðµâ²? Explain.
The ðð-coordinate of point ðšðšâ² is the same as the length of ðšðšâ²ðªðªâ²ï¿œï¿œï¿œï¿œï¿œï¿œ. Since |ðšðšâ²ðªðªâ²| = ðð|ðšðšðªðª|, then |ðšðšâ²ðªðªâ²| = ðð
ððâ ðð = ðððð
ðð. The location of point ðšðšâ² is ï¿œðð, ðððð
ððï¿œ, or approximately (ðð,ðð.ðð). The ðð-coordinate of
point ð©ð©â² is the same as the length of ð©ð©â²ðªðªâ²ï¿œï¿œï¿œï¿œï¿œï¿œ. Since |ð©ð©â²ðªðªâ²| = ðð|ð©ð©ðªðª|, then |ð©ð©â²ðªðªâ²| = ðð
ððâ ðð = ðð
ðð. The location
of point ð©ð©â² is ï¿œðð, ððððï¿œ, or approximately (ðð,ðð.ðð).
This is just like the last problem, but now I have to find coordinates for two points. Iâll need to mark a point ð¶ð¶ on the ð¥ð¥-axis on the same vertical line as points ðŽðŽ and ðµðµ.
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Lesson 6: Dilations on the Coordinate Plane
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G8-M3-Lesson 6: Dilations on the Coordinate Plane
1. Triangle ðŽðŽðŽðŽðŽðŽ is shown on the coordinate plane below. The triangle is dilated from the origin by scale factor ðð = 2. Identify the coordinates of the dilated triangle ðŽðŽâ²ðŽðŽâ²ðŽðŽâ².
ðšðš(âðð,âðð) â ðšðšâ²ï¿œðð â (âðð),ðð â (âðð)ï¿œ = ðšðšâ²(âðð,âðð) ð©ð©(âðð,ðð) â ð©ð©â²(ðð â (âðð),ðð â ðð) = ð©ð©â²(âðð,ðð) ðªðª(ðð,ðð) â ðªðªâ²(ðð â ðð,ðð â ðð) = ðªðªâ²(ðð,ðð)
The coordinates of the dilated triangle will be ðšðšâ²(âðð,âðð), ð©ð©â²(âðð,ðð), ðªðªâ²(ðð,ðð).
2. The figure ðŽðŽðŽðŽðŽðŽðŽðŽ has coordinates ðŽðŽ(3, 1), ðŽðŽ(12, 9), ðŽðŽ(â9, 3), and ðŽðŽ(â12,â3). The figure is dilated
from the origin by a scale factor ðð = 23. Identify the coordinates of the dilated figure ðŽðŽâ²ðŽðŽâ²ðŽðŽâ²ðŽðŽâ².
ðšðš(ðð,ðð) â ðšðšâ² ï¿œððððâ ðð,
ððððâ ððï¿œ = ðšðšâ² ï¿œðð,
ððððï¿œ
ð©ð©(ðððð,ðð) â ð©ð©â² ï¿œððððâ ðððð,
ððððâ ððï¿œ = ð©ð©â²(ðð,ðð)
ðªðª(âðð,ðð) â ðªðªâ² ï¿œððððâ (âðð),
ððððâ ððï¿œ = ðªðªâ²(âðð,ðð)
ð«ð«(âðððð,âðð) â ð«ð«â² ï¿œððððâ (âðððð),
ððððâ (âðð)ï¿œ = ð«ð«â²(âðð,âðð)
The coordinates of the dilated figure are ðšðšâ² ï¿œðð, ððððï¿œ, ð©ð©â² (ðð,ðð), ðªðªâ²(âðð, ðð), and ð«ð«â²(âðð,âðð).
The work we did in class led us to the conclusion that when given a point ðŽðŽ(ð¥ð¥,ðŠðŠ), we can find the coordinates of ðŽðŽâ² using the scale factor: ðŽðŽâ²(ððð¥ð¥, ðððŠðŠ). This only works for dilations from the origin.
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Lesson 7: Informal Proofs of Properties of Dilation
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G8-M3-Lesson 7: Informal Proofs of Properties of Dilation
1. A dilation from center ðð by scale factor ðð of an angle maps to what? Verify your claim on the coordinate plane.
The dilation of an angle maps to an angle.
To verify, I can choose any three points to create the angle and any scale factor. I can use what I learned in the last lesson to find the coordinates of the dilated points.
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Lesson 7: Informal Proofs of Properties of Dilation
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2. Prove the theorem: A dilation maps rays to rays. Let there be a dilation from center ðð with scale factor ðð so that ððâ² = ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð) and ððâ² = ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð). Show that ray ðððð maps to ray ððâ²ððâ² (i.e., that dilations map rays to rays). Using the diagram, answer the questions that follow, and then informally prove the theorem. (Hint: This proof is a lot like the proof for segments. This time, let ðð be a point on line ðððð that is not between points ðð and ðð.)
a. ðð is a point on ðððð.ᅵᅵᅵᅵᅵᅵâ By definition of dilation what is the name of ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·ð·(ðð)?
By the definition of dilation, we know that ðŒðŒâ² = ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«ð«(ðŒðŒ).
b. By the definition of dilation we know that ï¿œð¶ð¶ð·ð·â²ï¿œ
|ð¶ð¶ð·ð·| = ðð. What other two ratios are also equal to ðð?
By the definition of dilation, we know that ï¿œð¶ð¶ðžðžâ²ï¿œ
|ð¶ð¶ðžðž| = ï¿œð¶ð¶ðŒðŒâ²ï¿œ|ð¶ð¶ðŒðŒ| = ðð.
c. By FTS, what do we know about line ðððð and line ððâ²ððâ²?
By FTS, we know that line ð·ð·ðžðž and line ð·ð·â²ðžðžâ² are parallel.
d. What does FTS tell us about line ðððð and line ððâ²ððâ²?
By FTS, we know that line ðžðžðŒðŒ and line ðžðžâ²ðŒðŒâ² are parallel.
e. What conclusion can be drawn about the line that contains ððððᅵᅵᅵᅵᅵâ and the line ððâ²ððâ²?
The line that contains ð·ð·ðžðžï¿œï¿œï¿œï¿œï¿œï¿œâ is parallel to ðžðžâ²ðŒðŒâ²ï¿œâᅵᅵᅵᅵᅵᅵâ because ðžðžâ²ðŒðŒâ²ï¿œâᅵᅵᅵᅵᅵᅵâ is contained in the line ð·ð·â²ðžðžâ².
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Lesson 7: Informal Proofs of Properties of Dilation
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f. Informally prove that ððâ²ððâ²ï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œâ is a dilation of ðððð.ᅵᅵᅵᅵᅵᅵâ
Using the information from parts (a)â(e), we know that ðŒðŒ is a point on ð·ð·ðžðž.ᅵᅵᅵᅵᅵᅵᅵâ We also know that the line that contains ð·ð·ðžðžï¿œï¿œï¿œï¿œï¿œï¿œâ is parallel to line ðžðžâ²ðŒðŒâ². But we already know that ð·ð·ðžðžï¿œâᅵᅵᅵâ is parallel to ð·ð·â²ðžðžâ²ï¿œâᅵᅵᅵᅵᅵᅵâ . Since there can only be one line that passes through ðžðžâ² that is parallel to ð·ð·ðžðžï¿œï¿œï¿œï¿œï¿œï¿œâ , then the line that contains ð·ð·â²ðžðžâ²ï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œâ and line ðžðžâ²ðŒðŒâ² must coincide. That places the dilation of point ðŒðŒ, ðŒðŒâ², on ð·ð·â²ðžðžâ²ï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œï¿œâ , which proves that dilations map rays to rays.
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Lesson 8: Similarity
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G8-M3-Lesson 8: Similarity
1. In the picture below, we have a triangle ðŽðŽðŽðŽðŽðŽ that has been dilated from center ðð by scale factor ðð = 3. It is noted by ðŽðŽâ²ðŽðŽâ²ðŽðŽâ². We also have a triangle ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â², which is congruent to triangle ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² (i.e., â³ ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² â â³ ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â²). Describe the sequence of a dilation, followed by a congruence (of one or more rigid motions), that would map triangle ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â² onto triangle ðŽðŽðŽðŽðŽðŽ.
First, we must dilate triangle ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â² from center ðð by scale factor ðð = 1
3 to shrink it to the size of triangle ðŽðŽðŽðŽðŽðŽ. I will note this triangle
as ðŽðŽâ²â²â²ðŽðŽâ²â²â²ðŽðŽâ²â²â². Once I have the triangle to the right size, I can translate the dilated triangle, ðŽðŽâ²â²â²ðŽðŽâ²â²â²ðŽðŽâ²â²â², up one unit and to the left three units.
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Lesson 8: Similarity
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First, we must dilate triangle ðšðšâ²â²ð©ð©â²â²ðªðªâ²â² from center ð¶ð¶ by scale factor ðð = ðð
ðð to shrink it to the size of
triangle ðšðšð©ð©ðªðª. Next, we must translate the dilated triangle, noted by ðšðšâ²â²â²ð©ð©â²â²â²ðªðªâ²â²â², one unit up and three units to the left. This sequence of the dilation followed by the translation would map triangle ðšðšâ²â²ð©ð©â²â²ðªðªâ²â² onto triangle ðšðšð©ð©ðªðª.
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Lesson 8: Similarity
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2. Triangle ðŽðŽðŽðŽðŽðŽ is similar to triangle ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² (i.e., â³ ðŽðŽðŽðŽðŽðŽ ~ â³ ðŽðŽâ²ðŽðŽâ²ðŽðŽâ²). Prove the similarity by describing a sequence that would map triangle ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² onto triangle ðŽðŽðŽðŽðŽðŽ.
The scale factor that would magnify triangle ðšðšâ²ð©ð©â²ðªðªâ² to the size of triangle ðšðšð©ð©ðªðª is ðð = ðððð.
Sample description:
The sequence that would prove the similarity of the triangles is a dilation from a center by a scale factor of ðð = ðð
ðð, followed by a
translation along vector ðšðšâ²ðšðšï¿œï¿œï¿œï¿œï¿œï¿œï¿œâ , and finally, a rotation about point ðšðš.
I can check the ratios of the corresponding sides to see if they are the same proportion and equal to the same scale factor.
Once the triangle ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² is the same size as triangle ðŽðŽðŽðŽðŽðŽ, I can describe a congruence to map triangle ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² onto triangle ðŽðŽðŽðŽðŽðŽ.
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Lesson 9: Basic Properties of Similarity
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G8-M3-Lesson 9: Basic Properties of Similarity
1. In the diagram below, â³ ðŽðŽðŽðŽðŽðŽ ⌠ⳠðŽðŽâ²ðŽðŽâ²ðŽðŽâ² and â³ ðŽðŽâ²ðŽðŽâ²ðŽðŽâ² ~ â³ ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â². Is â³ ðŽðŽðŽðŽðŽðŽ ~ â³ ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â²? If so, describe the dilation followed by the congruence that demonstrates the similarity.
Yes, â³ ðšðšðšðšðšðš~â³ ðšðšâ²â²ðšðšâ²â²ðšðšâ²â² because similarity is transitive. Since ðð|ðšðšðšðš| = |ðšðšâ²â²ðšðšâ²â²|, |ðšðšðšðš| = ðð, and |ðšðšâ²â²ðšðšâ²â²| = ðð, then ðð â ðð = ðð. Therefore, ðð = ðð. Then, a dilation from the origin by scale factor ðð = ðð makes â³ðšðšðšðšðšðš the same size as â³ ðšðšâ²â²ðšðšâ²â²ðšðšâ²â². Translate the dilated image of â³ðšðšðšðšðšðš, â³ ðšðšâ²â²â²ðšðšâ²â²â²ðšðšâ²â²â², ðððð units to the right and ðð units up to map ðšðšâ²â²â² to ðšðšâ²â². Next, rotate the dilated image about point ðšðšâ²â², ðððð degrees clockwise. Finally, reflect the rotated image across line ðšðšâ²â²ðšðšâ²â². The sequence of the dilation and the congruence map â³ ðšðšðšðšðšðš onto â³ðšðšâ²â²ðšðšâ²â²ðšðšâ²â², demonstrating the similarity.
First I need to dilate â³ ðŽðŽðŽðŽðŽðŽ to the same size as â³ ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â². A dilation from the origin by ðð = 2 will make them the same size. Then I can describe a congruence to map the dilated image onto â³ ðŽðŽâ²â²ðŽðŽâ²â²ðŽðŽâ²â².
I remember that when I have to translate an image, it is best to do it so that corresponding points, like ðŽðŽ and ðŽðŽâ²â², coincide.
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Lesson 10: Informal Proof of AA Criterion for Similarity
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G8-M3-Lesson 10: Informal Proof of AA Criterion for Similarity
1. Are the triangles shown below similar? Present an informal argument as to why they are or are not similar.
Yes, â³ ðšðšðšðšðšðš ~ â³ ðšðšâ²ðšðšâ²ðšðšâ². They are similar because they have two pairs of corresponding angles that are equal in measure, namely, ï¿œâ ðšðšï¿œ = ï¿œâ ðšðšâ²ï¿œ = ðððððð°, and |â ðšðš| = |â ðšðšâ²| = ðððð°.
2. Are the triangles shown below similar? Present an informal argument as to why they are or are not similar.
No, â³ ðšðšðšðšðšðš is not similar to â³ ðšðšâ²ðšðšâ²ðšðšâ². By the given information, ï¿œâ ðšðšï¿œ â ï¿œâ ðšðšâ²ï¿œ, and ï¿œâ ðšðšï¿œ â ï¿œâ ðšðšâ²ï¿œ.
All I need for two triangles to be similar is for two corresponding angles to be equal in measure.
I can see that I have only one pair of corresponding angles, â ð¶ð¶ and â ð¶ð¶â², that are equal in measure.
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Lesson 10: Informal Proof of AA Criterion for Similarity
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3. Are the triangles shown below similar? Present an informal argument as to why they are or are not similar.
We do not know if â³ ðšðšðšðšðšðš is similar to â³ ðšðšâ²ðšðšâ²ðšðšâ². We can use the triangle sum theorem to find out that ï¿œâ ðšðšâ²ï¿œ = ðððð°, but we do not have any information about ï¿œâ ðšðšï¿œ or ï¿œâ ðšðšï¿œ. To be considered similar, the two triangles must have two pairs of corresponding angles that are equal in measure. In this problem, we only know the measures of one pair of corresponding angles.
I can use the triangle sum theorem to find the measure of â ðµðµâ².
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Lesson 11: More About Similar Triangles
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G8-M3-Lesson 11: More About Similar Triangles
1. In the diagram below, you have â³ ðŽðŽðŽðŽðŽðŽ and â³ ðŽðŽâ²ðŽðŽâ²ðŽðŽâ². Use this information to answer parts (a)â(b).
a. Based on the information given, is â³ ðŽðŽðŽðŽðŽðŽ ~ â³ ðŽðŽâ²ðŽðŽâ²ðŽðŽâ²? Explain.
Yes, â³ ðšðšðšðšðšðš ~ â³ ðšðšâ²ðšðšâ²ðšðšâ². Since there is only information about one pair of corresponding angles being equal in measure, then the corresponding sides must be checked to see if their ratios are equal.
ðð.ðððððð.ðððð= ðð.ðð
ðð.ðð
ðð.ðððððð⊠= ðð.ððððððâŠ
Since the values of these ratios are equal, approximately ðð.ðððððð, the triangles are similar.
b. Assume the length of side ðŽðŽðŽðŽï¿œï¿œï¿œï¿œ is 4.03. What is the length of side ðŽðŽâ²ðŽðŽâ²ï¿œï¿œï¿œï¿œï¿œï¿œ?
Let ðð represent the length of side ðšðšâ²ðšðšâ²ï¿œï¿œï¿œï¿œï¿œï¿œ. ðð
ðð.ðððð= ðð.ðððð.ðððð
We are looking for the value of ðð that makes the fractions equivalent. Therefore, ðð.ðððððð = ðððð.ðððððð, and ðð = ðð.ðð. The length of side ðšðšâ²ðšðšâ²ï¿œï¿œï¿œï¿œï¿œï¿œ is ðð.ðð.
I donât have enough information to use the AA criterion. I need to check the ratios of corresponding sides to see if they are equal.
After I set up the ratio, I need to find the value of ð¥ð¥ that makes the fractions equivalent.
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Lesson 11: More About Similar Triangles
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2. In the diagram below, you have â³ ðŽðŽðŽðŽðŽðŽ and â³ ðŽðŽâ²ðŽðŽðŽðŽâ². Based on the information given, is â³ ðŽðŽðŽðŽðŽðŽ ~ â³ ðŽðŽâ²ðŽðŽðŽðŽâ²? Explain.
Since both triangles have a common vertex, then |â ðšðš| = |â ðšðš|. This means that the measure of â ðšðš in â³ ðšðšðšðšðšðš is equal to the measure of â ðšðš in â³ ðšðšâ²ðšðšðšðšâ². However, there is not enough information provided to determine if the triangles are similar. We would need information about a pair of corresponding angles or more information about the side lengths of each of the triangles.
Although lines ðŽðŽðŽðŽ and ðŽðŽâ²ðŽðŽâ² look like they might be parallel, I donât know for sure. If they were parallel lines, I would have more information about corresponding angles of parallel lines.
I need to check the ratios of at least two sets of corresponding sides.
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Lesson 12: Modeling Using Similarity
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G8-M3-Lesson 12: Modeling Using Similarity
1. There is a statue of your schoolâs mascot in front of your school. You want to find out how tall the statue is, but it is too tall to measure directly. The following diagram represents the situation.
Describe the triangles in the situation, and explain how you know whether or not they are similar.
There are two triangles in the diagram, one formed by the statue and the shadow it casts, â³ ð¬ð¬ð¬ð¬ð¬ð¬, and another formed by the person and his shadow, â³ð©ð©ð©ð©ð¬ð¬. The triangles are similar if the height of the statue is measured at a ðððð° angle with the ground and if the person standing forms a ðððð° angle with the ground. We know that â ð©ð©ð¬ð¬ð©ð© is an angle common to both triangles. If |â ð¬ð¬ð¬ð¬ð¬ð¬| = |â ð©ð©ð©ð©ð¬ð¬| = ðððð°, then â³ ð¬ð¬ð¬ð¬ð¬ð¬ ~ â³ ð©ð©ð©ð©ð¬ð¬ by the AA criterion.
2. Assume â³ ðžðžðžðžðžðž ~ â³ ðµðµðµðµðžðž. If the statue casts a shadow 18 feet long and you are 5 feet tall and cast a shadow of 7 feet, find the height of the statue.
Let ðð represent the height of the statue; then ðððð = ðððð
ðð .
We are looking for the value of ðð that makes the fractions equivalent. Therefore, ðððð = ðððð, and ðð = ðððð
ðð. The statue is about ðððð feet tall.
If the height of the statue and I are measured at a 90° angle with the ground, then I can explain why triangles are similar. If they are not, I do not have enough information to determine if the triangles are similar.
Since I know the triangles are similar, I can set up a ratio of corresponding sides.
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Lesson 13: Proof of the Pythagorean Theorem
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G8-M3-Lesson 13: Proof of the Pythagorean Theorem
Use the Pythagorean theorem to determine the unknown length of the right triangle.
1. Determine the length of side ðð in the triangle below.
2. Determine the length of side ðð in the triangle below.
I know the missing side is the hypotenuse because it is the longest side of the triangle and opposite the right angle.
ðððððð + ðððððð = ðððð ðððððð + ðððððð = ðððð
ðððððð = ðððð ðððð = ðð
The side lengths are a tenth of the side lengths in problem one. I can multiply the side lengths by ten to make them whole numbers. That will make the problem easier.
ðððð + ðððð = ðð.ðððð ðð + ðððð = ðð.ðððð
ðð â ðð + ðððð = ðð.ðððð â ðð ðððð = ðð.ðððð
ðð = ðð.ðð
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Lesson 14: The Converse of the Pythagorean Theorem
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G8-M3-Lesson 14: The Converse of the Pythagorean Theorem
1. The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown below a right triangle? Show your work, and answer in a complete sentence.
We need to check if ðððððð + ðððððð = ðððððð is a true statement. The left side of the equation is equal to ðð,ðððððð. The right side of the equation is equal to ðð,ðððððð. That means ðððððð + ðððððð = ðððððð is true, and the triangle shown is a right triangle.
2. The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown below a right triangle? Show your work, and answer in a complete sentence.
We need to check if ðððððð + ðððððð = ðððððð is a true statement. The left side of the equation is equal to ðð,ðððððð. The right side of the equation is equal to ðð,ðððððð. That means ðððððð + ðððððð = ðððððð is not true, and the triangle shown is not a right triangle.
I remember that ðð is the longest side of the triangle.
I need to check to see if ðð2 + ðð2 = ðð2 is true. If it is true, then it is right triangle.
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