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Eureka Math, A Story of Ratiosยฎ
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Copyright ยฉ 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold, or commercialized, in whole or in part, without consent of the copyright holder. Please see our User Agreement for more information. โGreat Mindsโ and โEureka Mathโ are registered trademarks of Great Minds.
Eureka Mathโข Homework Helper
2015โ2016
Grade 8 Module 3
Lessons 1โ14
2015-16
Lesson 1: What Lies Behind โSame Shapeโ?
8โข3
G8-M3-Lesson 1: What Lies Behind โSame Shapeโ?
1. Let there be a dilation from center ๐๐. Then, ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐) = ๐๐โฒ, and ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐) = ๐๐โฒ. Examine thedrawing below. What can you determine about the scale factor of the dilation?
The dilation must have a scale factor larger than ๐๐, ๐๐ > ๐๐, since the dilated points are farther from the center than the original points.
2. Let there be a dilation from center ๐๐ with a scale factor ๐๐ = 2. Then, ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐) = ๐๐โฒ, and๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐) = ๐๐โฒ. |๐๐๐๐| = 1.7 cm, and |๐๐๐๐| = 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 |๐๐๐๐โฒ|.|๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐|๐ถ๐ถ๐ถ๐ถ|; therefore, |๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐ โ (๐๐.๐๐) = ๐๐.๐๐ and |๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐.๐๐ ๐๐๐๐.
b. Use the definition of dilation to determine |๐๐๐๐โฒ|.|๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐|๐ถ๐ถ๐ถ๐ถ|; therefore, |๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐ โ (๐๐.๐๐) = ๐๐.๐๐ and |๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐.๐๐ ๐๐๐๐.
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, |๐๐๐๐โฒ| is said, โThe length of segment ๐๐๐๐ prime.โ
We talked about the definition of dilation in class today. I should check the Lesson Summary box to review the definition.
1
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 1: What Lies Behind โSame Shapeโ?
8โข3
3. Let there be a dilation from center ๐๐ with a scale factor ๐๐. Then, ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐ต๐ต) = ๐ต๐ตโฒ, ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐ถ๐ถ) = ๐ถ๐ถโฒ, and |๐๐๐ต๐ต| = 10.8 cm, |๐๐๐ถ๐ถ| = 5 cm, and |๐๐๐ต๐ตโฒ| = 2.7 cm, as shown. Use the drawing below to answer parts (a)โ(c).
a. Using the definition of dilation with |๐๐๐ต๐ต| and |๐๐๐ต๐ตโฒ|, determine the scale factor of the dilation.
|๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐|๐ถ๐ถ๐ถ๐ถ|, which means ๐๐.๐๐ = ๐๐ โ (๐๐๐๐.๐๐), then
๐๐.๐๐๐๐๐๐.๐๐
= ๐๐ ๐๐๐๐
= ๐๐
b. Use the definition of dilation to determine |๐๐๐ถ๐ถโฒ|.
Since the scale factor, ๐๐, is ๐๐๐๐, then |๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐
๐๐|๐ถ๐ถ๐ถ๐ถ|;
therefore, |๐ถ๐ถ๐ถ๐ถโฒ| = ๐๐
๐๐โ ๐๐ = ๐๐.๐๐๐๐, and |๐ถ๐ถ๐ถ๐ถโฒ| =
๐๐.๐๐๐๐ ๐๐๐๐.
Since |๐๐๐ต๐ตโฒ| = ๐๐|๐๐๐ต๐ต|, then by the multiplication property of
equality, ๏ฟฝ๐๐๐ต๐ตโฒ๏ฟฝ
|๐๐๐ต๐ต| = ๐๐.
2
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 2: Properties of Dilations
8โข3
G8-M3-Lesson 2: Properties of Dilations
1. Given center ๐๐ and quadrilateral ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด, use a ruler to dilate the figure from center ๐๐ by a scale factor of ๐๐ = 1
4. Label the dilated quadrilateral ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ.
The figure in red below shows the dilated image of ๐จ๐จ๐จ๐จ๐จ๐จ๐จ๐จ. All measurements are in centimeters.
|๐ถ๐ถ๐จ๐จโฒ| = ๐๐|๐ถ๐ถ๐จ๐จ| |๐ถ๐ถ๐จ๐จโฒ| = ๐๐|๐ถ๐ถ๐จ๐จ| |๐ถ๐ถ๐จ๐จโฒ| = ๐๐|๐ถ๐ถ๐จ๐จ| |๐ถ๐ถ๐จ๐จโฒ| = ๐๐|๐ถ๐ถ๐จ๐จ|
|๐ถ๐ถ๐จ๐จโฒ| = ๐๐๐๐
(๐๐๐๐.๐๐) |๐ถ๐ถ๐จ๐จโฒ| = ๐๐๐๐
(๐๐.๐๐) |๐ถ๐ถ๐จ๐จโฒ| = ๐๐๐๐
(๐๐.๐๐) |๐ถ๐ถ๐จ๐จโฒ| = ๐๐๐๐
(๐๐.๐๐)
|๐ถ๐ถ๐จ๐จโฒ| = ๐๐.๐๐ |๐ถ๐ถ๐จ๐จโฒ| = ๐๐.๐๐ |๐ถ๐ถ๐จ๐จโฒ| = ๐๐.๐๐ |๐ถ๐ถ๐จ๐จโฒ| = ๐๐.๐๐
I need to draw the rays from the center ๐๐ to each point on the figure and then measure the length from ๐๐ 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.
3
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 2: Properties of Dilations
8โข3
2. Use a compass to dilate the figure ๐ด๐ด๐ด๐ด๐ด๐ด from center ๐๐, with scale factor ๐๐ = 3.
The figure in red below shows the dilated image of ๐จ๐จ๐จ๐จ๐จ๐จ.
I need to draw the rays like before, but this time I can use a compass to measure the distance from the center ๐๐ to a point and again use the compass to find the length 3 times from the center.
4
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 2: Properties of Dilations
8โข3
3. Use a compass to dilate the figure ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด from center ๐๐, with scale factor ๐๐ = 2. a. Dilate the same figure, ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด, from a new center, ๐๐โฒ, with scale factor ๐๐ = 2. Use double primes
(๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ) to distinguish this image from the original.
The figure in blue, ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ, shows the dilation of ๐จ๐จ๐จ๐จ๐จ๐จ๐จ๐จ from center ๐ถ๐ถ, with scale factor ๐๐ = ๐๐. The figure in red, ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ, shows the dilation of ๐จ๐จ๐จ๐จ๐จ๐จ๐จ๐จ from center ๐ถ๐ถโฒ with scale factor ๐๐ = ๐๐.
Since the original image was dilated by a scale factor of 2 each time, I know that the dilated figures are congruent.
5
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 2: Properties of Dilations
8โข3
b. What rigid motion, or sequence of rigid motions, would map ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ to ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ?
A translation along vector ๐จ๐จโฒโฒ๐จ๐จโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ (or any vector that connects a point of ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ and its corresponding point of ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ) would map the figure ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ to ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ.
The image below (with rays removed for clarity) shows the vector ๐จ๐จโฒโฒ๐จ๐จโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ .
4. A line segment ๐ด๐ด๐ด๐ด undergoes a dilation. Based on todayโs lesson, what is the image of the segment?
The segment dilates as a segment.
5. โ ๐ด๐ด๐๐๐ด๐ด measures 24ยฐ. After a dilation, what is the measure of โ ๐ด๐ดโฒ๐๐๐ด๐ดโฒ? How do you know?
The measure of โ ๐จ๐จโฒ๐ถ๐ถ๐จ๐จโฒ is ๐๐๐๐ยฐ. Dilations preserve angle measure, so โ ๐จ๐จโฒ๐ถ๐ถ๐จ๐จโฒ remains the same measure as โ ๐จ๐จ๐ถ๐ถ๐จ๐จ.
6
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 3: Examples of Dilations
8โข3
G8-M3-Lesson 3: Examples of Dilations
1. Dilate the figure from center ๐๐ by a scale factor of ๐๐ = 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 ๐ต๐ต, ๐ธ๐ธ, and ๐ท๐ท, then the dilated figure will look like a triangle. I will need to dilate more points along the curve ๐ต๐ต๐ท๐ท.
7
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 3: Examples of Dilations
8โข3
2. A triangle ๐ด๐ด๐ต๐ต๐ด๐ด was dilated from center ๐๐ by a scale factor of ๐๐ = 8. What scale factor would shrink the dilated figure back to the original size?
A scale factor of ๐๐ = ๐๐
๐๐ would bring the dilated figure
back to its original size.
3. A figure has been dilated from center ๐๐ by a scale factor of ๐๐ = 23. What scale factor would shrink the
dilated figure back to the original size?
A scale factor of ๐๐ = ๐๐
๐๐ 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 ๐๐ โ 1
๐๐= 1.
8
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 4: Fundamental Theorem of Similarity (FTS)
8โข3
G8-M3-Lesson 4: Fundamental Theorem of Similarity (FTS)
1. In the diagram below, points ๐๐, ๐๐, and ๐ด๐ด have been dilated from center ๐๐ by a scale factor of ๐๐ = 1
5 on a piece of lined paper.
a. What is the relationship between segments ๐๐๐๐ and ๐๐โฒ๐๐โฒ? How do you know?
๐ท๐ท๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ท๐ทโฒ๐ท๐ทโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are parallel because they follow the lines on the lined paper.
b. What is the relationship between segments ๐๐๐ด๐ด and ๐๐โฒ๐ด๐ดโฒ? How do you know?
๐ท๐ท๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ท๐ทโฒ๐ท๐ทโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are also parallel because they follow the lines on the lined paper.
c. Identify two angles whose measures are equal. How do you know?
โ ๐ถ๐ถ๐ท๐ท๐ท๐ท and โ ๐ถ๐ถ๐ท๐ทโฒ๐ท๐ทโฒ are equal in measure because they are corresponding angles created by parallel lines ๐ท๐ท๐ท๐ท and ๐ท๐ทโฒ๐ท๐ทโฒ.
d. What is the relationship between the lengths of segments ๐ด๐ด๐๐ and ๐ด๐ดโฒ๐๐โฒ? How do you know?
The length of segment ๐ท๐ทโฒ๐ท๐ทโฒ will be ๐๐๐๐ the length of segment ๐ท๐ท๐ท๐ท. The FTS states that the length of
the dilated segment is equal to the scale factor multiplied by the original segment length, or |๐ท๐ทโฒ๐ท๐ทโฒ| = ๐๐|๐ท๐ท๐ท๐ท|.
9
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 4: Fundamental Theorem of Similarity (FTS)
8โข3
2. Reynaldo sketched the following diagram on graph paper. He dilated points ๐ต๐ต and ๐ถ๐ถ from center ๐๐.
a. What is the scale factor ๐๐? Show your work.
|๐ถ๐ถ๐ฉ๐ฉโฒ| = ๐๐|๐ถ๐ถ๐ฉ๐ฉ| ๐๐ = ๐๐(๐๐) ๐๐๐๐
= ๐๐
๐๐ = ๐๐
b. Verify the scale factor with a different set of segments.
|๐ฉ๐ฉโฒ๐ช๐ชโฒ| = ๐๐|๐ฉ๐ฉ๐ช๐ช| ๐๐ = ๐๐(๐๐) ๐๐๐๐
= ๐๐
๐๐ = ๐๐
c. Which segments are parallel? How do you know?
Segments ๐ฉ๐ฉ๐ช๐ช and ๐ฉ๐ฉโฒ๐ช๐ชโฒ 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?
|โ ๐ถ๐ถ๐ฉ๐ฉโฒ๐ช๐ชโฒ| = |โ ๐ถ๐ถ๐ฉ๐ฉ๐ช๐ช|, and |โ ๐ถ๐ถ๐ช๐ชโฒ๐ฉ๐ฉโฒ| = |โ ๐ถ๐ถ๐ช๐ช๐ฉ๐ฉ| because they are corresponding angles of parallel lines cut by a transversal.
10
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 4: Fundamental Theorem of Similarity (FTS)
8โข3
3. Points ๐ต๐ต and ๐ถ๐ถ were dilated from center ๐๐.
a. What is the scale factor ๐๐? Show your work.
|๐ถ๐ถ๐ฉ๐ฉโฒ| = ๐๐|๐ถ๐ถ๐ฉ๐ฉ| ๐๐ = ๐๐(๐๐) ๐๐ = ๐๐
b. If |๐๐๐ถ๐ถ| โ 2.2, what is |๐๐๐ถ๐ถโฒ|?
|๐ถ๐ถ๐ช๐ชโฒ| = ๐๐|๐ถ๐ถ๐ช๐ช|
|๐ถ๐ถ๐ช๐ชโฒ| โ ๐๐(๐๐.๐๐)
|๐ถ๐ถ๐ช๐ชโฒ| โ ๐๐๐๐
c. How does the perimeter of โณ ๐๐๐ต๐ต๐ถ๐ถ compare to the perimeter of โณ ๐๐๐ต๐ตโฒ๐ถ๐ถโฒ?
Perimeter โณ๐ถ๐ถ๐ฉ๐ฉ๐ช๐ช โ ๐๐ + ๐๐ + ๐๐.๐๐ Perimeter โณ ๐ถ๐ถ๐ฉ๐ฉโฒ๐ช๐ชโฒ โ ๐๐ + ๐๐๐๐ + ๐๐๐๐
Perimeter โณ๐ถ๐ถ๐ฉ๐ฉ๐ช๐ช โ ๐๐.๐๐ Perimeter โณ ๐ถ๐ถ๐ฉ๐ฉโฒ๐ช๐ชโฒ โ ๐๐๐๐
d. Was the perimeter of โณ ๐๐๐ต๐ตโฒ๐ถ๐ถโฒ equal to the perimeter of โณ ๐๐๐ต๐ต๐ถ๐ถ multiplied by scale factor ๐๐? Explain.
Yes. The perimeter of โณ ๐ถ๐ถ๐ฉ๐ฉโฒ๐ช๐ชโฒ was five times the perimeter of โณ ๐ถ๐ถ๐ฉ๐ฉ๐ช๐ช, which makes sense because the dilation increased the length of each segment by a scale factor of ๐๐. That means that each side of โณ๐ถ๐ถ๐ฉ๐ฉโฒ๐ช๐ชโฒ was five times as long as each side of โณ๐ถ๐ถ๐ฉ๐ฉ๐ช๐ช.
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ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 5: First Consequences of FTS
8โข3
When we did this in class, I know we started by marking a point ๐ต๐ต on the ๐ฅ๐ฅ-axis on the same vertical line as point ๐ด๐ด. Then we used what we learned in the last lesson. I should review my classwork.
Now I know which vertical line ๐ด๐ดโฒ will fall on; it is the same as where ๐ต๐ตโฒ is.
G8-M3-Lesson 5: First Consequences of FTS
1. Dilate point ๐ด๐ด, located at (2, 4) from center ๐๐, by a scale factor of ๐๐ = 72. What is the precise location of
point ๐ด๐ดโฒ?
|๐ถ๐ถ๐ฉ๐ฉโฒ| = ๐๐|๐ถ๐ถ๐ฉ๐ฉ|
|๐ถ๐ถ๐ฉ๐ฉโฒ| =๐๐๐๐
(๐๐)
|๐ถ๐ถ๐ฉ๐ฉโฒ| = ๐๐
|๐จ๐จโฒ๐ฉ๐ฉโฒ| = ๐๐|๐จ๐จ๐ฉ๐ฉ|
|๐จ๐จโฒ๐ฉ๐ฉโฒ| =๐๐๐๐
(๐๐)
|๐จ๐จโฒ๐ฉ๐ฉโฒ| =๐๐๐๐๐๐
|๐จ๐จโฒ๐ฉ๐ฉโฒ| = ๐๐๐๐
I know that segment ๐ด๐ดโฒ๐ต๐ตโฒ is a dilation of segment ๐ด๐ด๐ต๐ต by the same scale factor ๐๐ = 7
2. Since ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is
contained within a vertical line, I can easily determine its length is 4 units.
Therefore, ๐จ๐จโฒ is located at (๐๐,๐๐๐๐).
12
ยฉ 2015 Great Minds eureka-math.org G8-M3-HWH-1.3.0-09.2015
Homework Helper A Story of Ratios
2015-16
Lesson 5: First Consequences of FTS
8โข3
2. Dilate point ๐ด๐ด, located at (7, 5) from center ๐๐, by a scale factor of ๐๐ = 37. Then, dilate point ๐ต๐ต, located at
(7, 3) from center ๐๐, by a scale factor of ๐๐ = 37. What are the coordinates of ๐ด๐ดโฒ and ๐ต๐ตโฒ? Explain.
The ๐๐-coordinate of point ๐จ๐จโฒ is the same as the length of ๐จ๐จโฒ๐ช๐ชโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Since |๐จ๐จโฒ๐ช๐ชโฒ| = ๐๐|๐จ๐จ๐ช๐ช|, then |๐จ๐จโฒ๐ช๐ชโฒ| = ๐๐
๐๐โ ๐๐ = ๐๐๐๐
๐๐. The location of point ๐จ๐จโฒ is ๏ฟฝ๐๐, ๐๐๐๐
๐๐๏ฟฝ, or approximately (๐๐,๐๐.๐๐). The ๐๐-coordinate of
point ๐ฉ๐ฉโฒ is the same as the length of ๐ฉ๐ฉโฒ๐ช๐ชโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Since |๐ฉ๐ฉโฒ๐ช๐ชโฒ| = ๐๐|๐ฉ๐ฉ๐ช๐ช|, then |๐ฉ๐ฉโฒ๐ช๐ชโฒ| = ๐๐
๐๐โ ๐๐ = ๐๐
๐๐. The location
of point ๐ฉ๐ฉโฒ is ๏ฟฝ๐๐, ๐๐๐๐๏ฟฝ, or approximately (๐๐,๐๐.๐๐).
This is just like the last problem, but now I have to find coordinates for two points. Iโll need to mark a point ๐ถ๐ถ on the ๐ฅ๐ฅ-axis on the same vertical line as points ๐ด๐ด and ๐ต๐ต.
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Lesson 6: Dilations on the Coordinate Plane
8โข3
G8-M3-Lesson 6: Dilations on the Coordinate Plane
1. Triangle ๐ด๐ด๐ด๐ด๐ด๐ด is shown on the coordinate plane below. The triangle is dilated from the origin by scale factor ๐๐ = 2. Identify the coordinates of the dilated triangle ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ.
๐จ๐จ(โ๐๐,โ๐๐) โ ๐จ๐จโฒ๏ฟฝ๐๐ โ (โ๐๐),๐๐ โ (โ๐๐)๏ฟฝ = ๐จ๐จโฒ(โ๐๐,โ๐๐) ๐ฉ๐ฉ(โ๐๐,๐๐) โ ๐ฉ๐ฉโฒ(๐๐ โ (โ๐๐),๐๐ โ ๐๐) = ๐ฉ๐ฉโฒ(โ๐๐,๐๐) ๐ช๐ช(๐๐,๐๐) โ ๐ช๐ชโฒ(๐๐ โ ๐๐,๐๐ โ ๐๐) = ๐ช๐ชโฒ(๐๐,๐๐)
The coordinates of the dilated triangle will be ๐จ๐จโฒ(โ๐๐,โ๐๐), ๐ฉ๐ฉโฒ(โ๐๐,๐๐), ๐ช๐ชโฒ(๐๐,๐๐).
2. The figure ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด has coordinates ๐ด๐ด(3, 1), ๐ด๐ด(12, 9), ๐ด๐ด(โ9, 3), and ๐ด๐ด(โ12,โ3). The figure is dilated
from the origin by a scale factor ๐๐ = 23. Identify the coordinates of the dilated figure ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ.
๐จ๐จ(๐๐,๐๐) โ ๐จ๐จโฒ ๏ฟฝ๐๐๐๐โ ๐๐,
๐๐๐๐โ ๐๐๏ฟฝ = ๐จ๐จโฒ ๏ฟฝ๐๐,
๐๐๐๐๏ฟฝ
๐ฉ๐ฉ(๐๐๐๐,๐๐) โ ๐ฉ๐ฉโฒ ๏ฟฝ๐๐๐๐โ ๐๐๐๐,
๐๐๐๐โ ๐๐๏ฟฝ = ๐ฉ๐ฉโฒ(๐๐,๐๐)
๐ช๐ช(โ๐๐,๐๐) โ ๐ช๐ชโฒ ๏ฟฝ๐๐๐๐โ (โ๐๐),
๐๐๐๐โ ๐๐๏ฟฝ = ๐ช๐ชโฒ(โ๐๐,๐๐)
๐ซ๐ซ(โ๐๐๐๐,โ๐๐) โ ๐ซ๐ซโฒ ๏ฟฝ๐๐๐๐โ (โ๐๐๐๐),
๐๐๐๐โ (โ๐๐)๏ฟฝ = ๐ซ๐ซโฒ(โ๐๐,โ๐๐)
The coordinates of the dilated figure are ๐จ๐จโฒ ๏ฟฝ๐๐, ๐๐๐๐๏ฟฝ, ๐ฉ๐ฉโฒ (๐๐,๐๐), ๐ช๐ชโฒ(โ๐๐, ๐๐), and ๐ซ๐ซโฒ(โ๐๐,โ๐๐).
The work we did in class led us to the conclusion that when given a point ๐ด๐ด(๐ฅ๐ฅ,๐ฆ๐ฆ), we can find the coordinates of ๐ด๐ดโฒ using the scale factor: ๐ด๐ดโฒ(๐๐๐ฅ๐ฅ, ๐๐๐ฆ๐ฆ). This only works for dilations from the origin.
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Lesson 7: Informal Proofs of Properties of Dilation
8โข3
G8-M3-Lesson 7: Informal Proofs of Properties of Dilation
1. A dilation from center ๐๐ by scale factor ๐๐ 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 ๐๐ with scale factor ๐๐ so that ๐๐โฒ = ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐) and ๐๐โฒ = ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐). Show that ray ๐๐๐๐ maps to ray ๐๐โฒ๐๐โฒ (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 ๐๐ be a point on line ๐๐๐๐ that is not between points ๐๐ and ๐๐.)
a. ๐๐ is a point on ๐๐๐๐.๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ By definition of dilation what is the name of ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท(๐๐)?
By the definition of dilation, we know that ๐ผ๐ผโฒ = ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ(๐ผ๐ผ).
b. By the definition of dilation we know that ๏ฟฝ๐ถ๐ถ๐ท๐ทโฒ๏ฟฝ
|๐ถ๐ถ๐ท๐ท| = ๐๐. What other two ratios are also equal to ๐๐?
By the definition of dilation, we know that ๏ฟฝ๐ถ๐ถ๐ธ๐ธโฒ๏ฟฝ
|๐ถ๐ถ๐ธ๐ธ| = ๏ฟฝ๐ถ๐ถ๐ผ๐ผโฒ๏ฟฝ|๐ถ๐ถ๐ผ๐ผ| = ๐๐.
c. By FTS, what do we know about line ๐๐๐๐ and line ๐๐โฒ๐๐โฒ?
By FTS, we know that line ๐ท๐ท๐ธ๐ธ and line ๐ท๐ทโฒ๐ธ๐ธโฒ are parallel.
d. What does FTS tell us about line ๐๐๐๐ and line ๐๐โฒ๐๐โฒ?
By FTS, we know that line ๐ธ๐ธ๐ผ๐ผ and line ๐ธ๐ธโฒ๐ผ๐ผโฒ are parallel.
e. What conclusion can be drawn about the line that contains ๐๐๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ and the line ๐๐โฒ๐๐โฒ?
The line that contains ๐ท๐ท๐ธ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ is parallel to ๐ธ๐ธโฒ๐ผ๐ผโฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ because ๐ธ๐ธโฒ๐ผ๐ผโฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ is contained in the line ๐ท๐ทโฒ๐ธ๐ธโฒ.
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Lesson 7: Informal Proofs of Properties of Dilation
8โข3
f. Informally prove that ๐๐โฒ๐๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ is a dilation of ๐๐๐๐.๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ
Using the information from parts (a)โ(e), we know that ๐ผ๐ผ is a point on ๐ท๐ท๐ธ๐ธ.๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ We also know that the line that contains ๐ท๐ท๐ธ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ is parallel to line ๐ธ๐ธโฒ๐ผ๐ผโฒ. But we already know that ๐ท๐ท๐ธ๐ธ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ is parallel to ๐ท๐ทโฒ๐ธ๐ธโฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ . Since there can only be one line that passes through ๐ธ๐ธโฒ that is parallel to ๐ท๐ท๐ธ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ , then the line that contains ๐ท๐ทโฒ๐ธ๐ธโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ and line ๐ธ๐ธโฒ๐ผ๐ผโฒ must coincide. That places the dilation of point ๐ผ๐ผ, ๐ผ๐ผโฒ, on ๐ท๐ทโฒ๐ธ๐ธโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ , which proves that dilations map rays to rays.
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Lesson 8: Similarity
8โข3
G8-M3-Lesson 8: Similarity
1. In the picture below, we have a triangle ๐ด๐ด๐ด๐ด๐ด๐ด that has been dilated from center ๐๐ by scale factor ๐๐ = 3. It is noted by ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ. We also have a triangle ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ, which is congruent to triangle ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ (i.e., โณ ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ โ โณ ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ). Describe the sequence of a dilation, followed by a congruence (of one or more rigid motions), that would map triangle ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ onto triangle ๐ด๐ด๐ด๐ด๐ด๐ด.
First, we must dilate triangle ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ from center ๐๐ by scale factor ๐๐ = 1
3 to shrink it to the size of triangle ๐ด๐ด๐ด๐ด๐ด๐ด. I will note this triangle
as ๐ด๐ดโฒโฒโฒ๐ด๐ดโฒโฒโฒ๐ด๐ดโฒโฒโฒ. Once I have the triangle to the right size, I can translate the dilated triangle, ๐ด๐ดโฒโฒโฒ๐ด๐ดโฒโฒโฒ๐ด๐ดโฒโฒโฒ, 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 ๐จ๐จโฒโฒ๐ฉ๐ฉโฒโฒ๐ช๐ชโฒโฒ from center ๐ถ๐ถ by scale factor ๐๐ = ๐๐
๐๐ to shrink it to the size of
triangle ๐จ๐จ๐ฉ๐ฉ๐ช๐ช. Next, we must translate the dilated triangle, noted by ๐จ๐จโฒโฒโฒ๐ฉ๐ฉโฒโฒโฒ๐ช๐ชโฒโฒโฒ, one unit up and three units to the left. This sequence of the dilation followed by the translation would map triangle ๐จ๐จโฒโฒ๐ฉ๐ฉโฒโฒ๐ช๐ชโฒโฒ onto triangle ๐จ๐จ๐ฉ๐ฉ๐ช๐ช.
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Lesson 8: Similarity
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2. Triangle ๐ด๐ด๐ด๐ด๐ด๐ด is similar to triangle ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ (i.e., โณ ๐ด๐ด๐ด๐ด๐ด๐ด ~ โณ ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ). Prove the similarity by describing a sequence that would map triangle ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ onto triangle ๐ด๐ด๐ด๐ด๐ด๐ด.
The scale factor that would magnify triangle ๐จ๐จโฒ๐ฉ๐ฉโฒ๐ช๐ชโฒ to the size of triangle ๐จ๐จ๐ฉ๐ฉ๐ช๐ช is ๐๐ = ๐๐๐๐.
Sample description:
The sequence that would prove the similarity of the triangles is a dilation from a center by a scale factor of ๐๐ = ๐๐
๐๐, followed by a
translation along vector ๐จ๐จโฒ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ , and finally, a rotation about point ๐จ๐จ.
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 ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ is the same size as triangle ๐ด๐ด๐ด๐ด๐ด๐ด, I can describe a congruence to map triangle ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ onto triangle ๐ด๐ด๐ด๐ด๐ด๐ด.
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Lesson 9: Basic Properties of Similarity
8โข3
G8-M3-Lesson 9: Basic Properties of Similarity
1. In the diagram below, โณ ๐ด๐ด๐ด๐ด๐ด๐ด โผ โณ ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ and โณ ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ ~ โณ ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ. Is โณ ๐ด๐ด๐ด๐ด๐ด๐ด ~ โณ ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ? If so, describe the dilation followed by the congruence that demonstrates the similarity.
Yes, โณ ๐จ๐จ๐จ๐จ๐จ๐จ~โณ ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ because similarity is transitive. Since ๐๐|๐จ๐จ๐จ๐จ| = |๐จ๐จโฒโฒ๐จ๐จโฒโฒ|, |๐จ๐จ๐จ๐จ| = ๐๐, and |๐จ๐จโฒโฒ๐จ๐จโฒโฒ| = ๐๐, then ๐๐ โ ๐๐ = ๐๐. Therefore, ๐๐ = ๐๐. Then, a dilation from the origin by scale factor ๐๐ = ๐๐ makes โณ๐จ๐จ๐จ๐จ๐จ๐จ the same size as โณ ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ. Translate the dilated image of โณ๐จ๐จ๐จ๐จ๐จ๐จ, โณ ๐จ๐จโฒโฒโฒ๐จ๐จโฒโฒโฒ๐จ๐จโฒโฒโฒ, ๐๐๐๐ units to the right and ๐๐ units up to map ๐จ๐จโฒโฒโฒ to ๐จ๐จโฒโฒ. Next, rotate the dilated image about point ๐จ๐จโฒโฒ, ๐๐๐๐ degrees clockwise. Finally, reflect the rotated image across line ๐จ๐จโฒโฒ๐จ๐จโฒโฒ. The sequence of the dilation and the congruence map โณ ๐จ๐จ๐จ๐จ๐จ๐จ onto โณ๐จ๐จโฒโฒ๐จ๐จโฒโฒ๐จ๐จโฒโฒ, demonstrating the similarity.
First I need to dilate โณ ๐ด๐ด๐ด๐ด๐ด๐ด to the same size as โณ ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ. A dilation from the origin by ๐๐ = 2 will make them the same size. Then I can describe a congruence to map the dilated image onto โณ ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ๐ด๐ดโฒโฒ.
I remember that when I have to translate an image, it is best to do it so that corresponding points, like ๐ด๐ด and ๐ด๐ดโฒโฒ, coincide.
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Lesson 10: Informal Proof of AA Criterion for Similarity
8โข3
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, โณ ๐จ๐จ๐จ๐จ๐จ๐จ ~ โณ ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ. They are similar because they have two pairs of corresponding angles that are equal in measure, namely, ๏ฟฝโ ๐จ๐จ๏ฟฝ = ๏ฟฝโ ๐จ๐จโฒ๏ฟฝ = ๐๐๐๐๐๐ยฐ, and |โ ๐จ๐จ| = |โ ๐จ๐จโฒ| = ๐๐๐๐ยฐ.
2. Are the triangles shown below similar? Present an informal argument as to why they are or are not similar.
No, โณ ๐จ๐จ๐จ๐จ๐จ๐จ is not similar to โณ ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ. By the given information, ๏ฟฝโ ๐จ๐จ๏ฟฝ โ ๏ฟฝโ ๐จ๐จโฒ๏ฟฝ, and ๏ฟฝโ ๐จ๐จ๏ฟฝ โ ๏ฟฝโ ๐จ๐จโฒ๏ฟฝ.
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, โ ๐ถ๐ถ and โ ๐ถ๐ถโฒ, 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 โณ ๐จ๐จ๐จ๐จ๐จ๐จ is similar to โณ ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ. We can use the triangle sum theorem to find out that ๏ฟฝโ ๐จ๐จโฒ๏ฟฝ = ๐๐๐๐ยฐ, but we do not have any information about ๏ฟฝโ ๐จ๐จ๏ฟฝ or ๏ฟฝโ ๐จ๐จ๏ฟฝ. 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 โ ๐ต๐ตโฒ.
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Lesson 11: More About Similar Triangles
8โข3
G8-M3-Lesson 11: More About Similar Triangles
1. In the diagram below, you have โณ ๐ด๐ด๐ด๐ด๐ด๐ด and โณ ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ. Use this information to answer parts (a)โ(b).
a. Based on the information given, is โณ ๐ด๐ด๐ด๐ด๐ด๐ด ~ โณ ๐ด๐ดโฒ๐ด๐ดโฒ๐ด๐ดโฒ? Explain.
Yes, โณ ๐จ๐จ๐จ๐จ๐จ๐จ ~ โณ ๐จ๐จโฒ๐จ๐จโฒ๐จ๐จโฒ. 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.
๐๐.๐๐๐๐๐๐.๐๐๐๐= ๐๐.๐๐
๐๐.๐๐
๐๐.๐๐๐๐๐๐โฆ = ๐๐.๐๐๐๐๐๐โฆ
Since the values of these ratios are equal, approximately ๐๐.๐๐๐๐๐๐, the triangles are similar.
b. Assume the length of side ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is 4.03. What is the length of side ๐ด๐ดโฒ๐ด๐ดโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ?
Let ๐๐ represent the length of side ๐จ๐จโฒ๐จ๐จโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. ๐๐
๐๐.๐๐๐๐= ๐๐.๐๐๐๐.๐๐๐๐
We are looking for the value of ๐๐ that makes the fractions equivalent. Therefore, ๐๐.๐๐๐๐๐๐ = ๐๐๐๐.๐๐๐๐๐๐, and ๐๐ = ๐๐.๐๐. The length of side ๐จ๐จโฒ๐จ๐จโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is ๐๐.๐๐.
I donโt 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 ๐ฅ๐ฅ that makes the fractions equivalent.
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Lesson 11: More About Similar Triangles
8โข3
2. In the diagram below, you have โณ ๐ด๐ด๐ด๐ด๐ด๐ด and โณ ๐ด๐ดโฒ๐ด๐ด๐ด๐ดโฒ. Based on the information given, is โณ ๐ด๐ด๐ด๐ด๐ด๐ด ~ โณ ๐ด๐ดโฒ๐ด๐ด๐ด๐ดโฒ? Explain.
Since both triangles have a common vertex, then |โ ๐จ๐จ| = |โ ๐จ๐จ|. This means that the measure of โ ๐จ๐จ in โณ ๐จ๐จ๐จ๐จ๐จ๐จ is equal to the measure of โ ๐จ๐จ in โณ ๐จ๐จโฒ๐จ๐จ๐จ๐จโฒ. 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 ๐ด๐ด๐ด๐ด and ๐ด๐ดโฒ๐ด๐ดโฒ look like they might be parallel, I donโt 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
8โข3
G8-M3-Lesson 12: Modeling Using Similarity
1. There is a statue of your schoolโs 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, โณ ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ, and another formed by the person and his shadow, โณ๐ฉ๐ฉ๐ฉ๐ฉ๐ฌ๐ฌ. The triangles are similar if the height of the statue is measured at a ๐๐๐๐ยฐ angle with the ground and if the person standing forms a ๐๐๐๐ยฐ angle with the ground. We know that โ ๐ฉ๐ฉ๐ฌ๐ฌ๐ฉ๐ฉ is an angle common to both triangles. If |โ ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ| = |โ ๐ฉ๐ฉ๐ฉ๐ฉ๐ฌ๐ฌ| = ๐๐๐๐ยฐ, then โณ ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ ~ โณ ๐ฉ๐ฉ๐ฉ๐ฉ๐ฌ๐ฌ by the AA criterion.
2. Assume โณ ๐ธ๐ธ๐ธ๐ธ๐ธ๐ธ ~ โณ ๐ต๐ต๐ต๐ต๐ธ๐ธ. 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 ๐๐ represent the height of the statue; then ๐๐๐๐ = ๐๐๐๐
๐๐ .
We are looking for the value of ๐๐ that makes the fractions equivalent. Therefore, ๐๐๐๐ = ๐๐๐๐, and ๐๐ = ๐๐๐๐
๐๐. The statue is about ๐๐๐๐ feet tall.
If the height of the statue and I are measured at a 90ยฐ 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
8โข3
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 ๐๐ in the triangle below.
2. Determine the length of side ๐๐ 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.
๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐
๐๐๐๐๐๐ = ๐๐๐๐ ๐๐๐๐ = ๐๐
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.
๐๐๐๐ + ๐๐๐๐ = ๐๐.๐๐๐๐ ๐๐ + ๐๐๐๐ = ๐๐.๐๐๐๐
๐๐ โ ๐๐ + ๐๐๐๐ = ๐๐.๐๐๐๐ โ ๐๐ ๐๐๐๐ = ๐๐.๐๐๐๐
๐๐ = ๐๐.๐๐
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Lesson 14: The Converse of the Pythagorean Theorem
8โข3
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 ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐๐๐ is a true statement. The left side of the equation is equal to ๐๐,๐๐๐๐๐๐. The right side of the equation is equal to ๐๐,๐๐๐๐๐๐. That means ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐๐๐ 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 ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐๐๐ is a true statement. The left side of the equation is equal to ๐๐,๐๐๐๐๐๐. The right side of the equation is equal to ๐๐,๐๐๐๐๐๐. That means ๐๐๐๐๐๐ + ๐๐๐๐๐๐ = ๐๐๐๐๐๐ is not true, and the triangle shown is not a right triangle.
I remember that ๐๐ is the longest side of the triangle.
I need to check to see if ๐๐2 + ๐๐2 = ๐๐2 is true. If it is true, then it is right triangle.
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