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Eureka Mathâą Homework Helper

2015â2016

Grade 8 Module 3

Lessons 1â14

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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 đđ. Then, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ, and đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ. Examine thedrawing below. What can you determine about the scale factor of the dilation?

The dilation must have a scale factor larger than đđ, đđ > đđ, since the dilated points are farther from the center than the original points.

2. Let there be a dilation from center đđ with a scale factor đđ = 2. Then, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ, andđ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ. |đđđđ| = 1.7 cm, and |đđđđ| = 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 |đđđđâČ|.|đ¶đ¶đ¶đ¶âČ| = đđ|đ¶đ¶đ¶đ¶|; therefore, |đ¶đ¶đ¶đ¶âČ| = đđ â (đđ.đđ) = đđ.đđ and |đ¶đ¶đ¶đ¶âČ| = đđ.đđ đđđđ.

b. Use the definition of dilation to determine |đđđđâČ|.|đ¶đ¶đ¶đ¶âČ| = đđ|đ¶đ¶đ¶đ¶|; therefore, |đ¶đ¶đ¶đ¶âČ| = đđ â (đđ.đđ) = đđ.đđ and |đ¶đ¶đ¶đ¶âČ| = đđ.đđ đđđđ.

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, |đđđđâČ| is said, âThe length of segment đđđđ 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 đđ with a scale factor đđ. Then, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đ”đ”) = đ”đ”âČ, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đ¶đ¶) = đ¶đ¶âČ, and |đđđ”đ”| = 10.8 cm, |đđđ¶đ¶| = 5 cm, and |đđđ”đ”âČ| = 2.7 cm, as shown. Use the drawing below to answer parts (a)â(c).

a. Using the definition of dilation with |đđđ”đ”| and |đđđ”đ”âČ|, determine the scale factor of the dilation.

|đ¶đ¶đ¶đ¶âČ| = đđ|đ¶đ¶đ¶đ¶|, which means đđ.đđ = đđ â (đđđđ.đđ), then

đđ.đđđđđđ.đđ

= đđ đđđđ

= đđ

b. Use the definition of dilation to determine |đđđ¶đ¶âČ|.

Since the scale factor, đđ, is đđđđ, then |đ¶đ¶đ¶đ¶âČ| = đđ

đđ|đ¶đ¶đ¶đ¶|;

therefore, |đ¶đ¶đ¶đ¶âČ| = đđđđâ đđ = đđ.đđđđ, and |đ¶đ¶đ¶đ¶âČ| =

đđ.đđđđ đđđđ.

Since |đđđ”đ”âČ| = đđ|đđđ”đ”|, then by the multiplication property of

equality, ïżœđđđ”đ”âČïżœ

|đđđ”đ”|= đđ.

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Lesson 2: Properties of Dilations

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G8-M3-Lesson 2: Properties of Dilations

1. Given center đđ and quadrilateral đŽđŽđŽđŽđŽđŽđŽđŽ, use a ruler to dilate the figure from center đđ by a scale factor of đđ = 1

4. Label the dilated quadrilateral đŽđŽâČđŽđŽâČđŽđŽâČđŽđŽâČ.

The figure in red below shows the dilated image of đšđšđšđšđšđšđšđš. All measurements are in centimeters.

|đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš| |đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš| |đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš| |đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš|

|đ¶đ¶đšđšâČ| = đđđđ

(đđđđ.đđ) |đ¶đ¶đšđšâČ| = đđđđ

(đđ.đđ) |đ¶đ¶đšđšâČ| = đđđđ

(đđ.đđ) |đ¶đ¶đšđšâČ| = đđđđ

(đđ.đđ)

|đ¶đ¶đšđšâČ| = đđ.đđ |đ¶đ¶đšđšâČ| = đđ.đđ |đ¶đ¶đšđšâČ| = đđ.đđ |đ¶đ¶đšđšâČ| = đđ.đđ

I need to draw the rays from the center đđ to each point on the figure and then measure the length from đđ 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 đŽđŽđŽđŽđŽđŽ from center đđ, with scale factor đđ = 3.

The figure in red below shows the dilated image of đšđšđšđšđšđš.

I need to draw the rays like before, but this time I can use a compass to measure the distance from the center đđ 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 đŽđŽđŽđŽđŽđŽđŽđŽ from center đđ, with scale factor đđ = 2. a. Dilate the same figure, đŽđŽđŽđŽđŽđŽđŽđŽ, from a new center, đđâČ, with scale factor đđ = 2. Use double primes

(đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ) to distinguish this image from the original.

The figure in blue, đšđšâČđšđšâČđšđšâČđšđšâČ, shows the dilation of đšđšđšđšđšđšđšđš from center đ¶đ¶, with scale factor đđ = đđ. The figure in red, đšđšâČâČđšđšâČâČđšđšâČâČđšđšâČâČ, shows the dilation of đšđšđšđšđšđšđšđš from center đ¶đ¶âČ with scale factor đđ = đđ.

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 đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ to đŽđŽâČđŽđŽâČđŽđŽâČđŽđŽâČ?

A translation along vector đšđšâČâČđšđšâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ (or any vector that connects a point of đšđšâČâČđšđšâČâČđšđšâČâČđšđšâČâČ and its corresponding point of đšđšâČđšđšâČđšđšâČđšđšâČ) would map the figure đšđšâČâČđšđšâČâČđšđšâČâČđšđšâČâČ to đšđšâČđšđšâČđšđšâČđšđšâČ.

The image below (with rays removed for clarity) shows the vector đšđšâČâČđšđšâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ .

4. A line segment đŽđŽđŽđŽ undergoes a dilation. Based on todayâs lesson, what is the image of the segment?

The segment dilates as a segment.

5. â đŽđŽđđđŽđŽ measures 24Â°. After a dilation, what is the measure of â đŽđŽâČđđđŽđŽâČ? How do you know?

The measure of â đšđšâČđ¶đ¶đšđšâČ is đđđđÂ°. Dilations preserve angle measure, so â đšđšâČđ¶đ¶đšđšâČ remains the same measure as â đšđšđ¶đ¶đšđš.

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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 đđ by a scale factor of đđ = 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 đ”đ”, đžđž, and đ·đ·, then the dilated figure will look like a triangle. I will need to dilate more points along the curve đ”đ”đ·đ·.

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Lesson 3: Examples of Dilations

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2. A triangle đŽđŽđ”đ”đŽđŽ was dilated from center đđ by a scale factor of đđ = 8. What scale factor would shrink the dilated figure back to the original size?

A scale factor of đđ = đđđđ would bring the dilated figure

back to its original size.

3. A figure has been dilated from center đđ by a scale factor of đđ = 23. What scale factor would shrink the dilated figure back to the original size?

A scale factor of đđ = đđđđ 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 đđ â 1

đđ= 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 đđ, đđ, and đŽđŽ have been dilated from center đđ by a scale factor of đđ = 1

5 on a piece of lined paper.

a. What is the relationship between segments đđđđ and đđâČđđâČ? How do you know?

đ·đ·đ·đ·ïżœïżœïżœïżœ and đ·đ·âČđ·đ·âČïżœïżœïżœïżœïżœïżœ are parallel because they follow the lines on the lined paper.

b. What is the relationship between segments đđđŽđŽ and đđâČđŽđŽâČ? How do you know?

đ·đ·đ·đ·ïżœïżœïżœïżœ and đ·đ·âČđ·đ·âČïżœïżœïżœïżœïżœïżœ 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 đ¶đ¶đ·đ·đ·đ· and â đ¶đ¶đ·đ·âČđ·đ·âČ are equal in measure because they are corresponding angles created by parallel lines đ·đ·đ·đ· and đ·đ·âČđ·đ·âČ.

d. What is the relationship between the lengths of segments đŽđŽđđ and đŽđŽâČđđâČ? How do you know?

The length of segment đ·đ·âČđ·đ·âČ will be đđđđ the length of segment đ·đ·đ·đ·. The FTS states that the length of

the dilated segment is equal to the scale factor multiplied by the original segment length, or |đ·đ·âČđ·đ·âČ| = đđ|đ·đ·đ·đ·|.

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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 đ”đ” and đ¶đ¶ from center đđ.

a. What is the scale factor đđ? Show your work.

|đ¶đ¶đ©đ©âČ| = đđ|đ¶đ¶đ©đ©| đđ = đđ(đđ) đđđđ

= đđ

đđ = đđ

b. Verify the scale factor with a different set of segments.

|đ©đ©âČđȘđȘâČ| = đđ|đ©đ©đȘđȘ| đđ = đđ(đđ) đđđđ

= đđ

đđ = đđ

c. Which segments are parallel? How do you know?

Segments đ©đ©đȘđȘ and đ©đ©âČđȘđȘâČ 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 đ¶đ¶đ©đ©âČđȘđȘâČ| = |â đ¶đ¶đ©đ©đȘđȘ|, and |â đ¶đ¶đȘđȘâČđ©đ©âČ| = |â đ¶đ¶đȘđȘđ©đ©| 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 đ”đ” and đ¶đ¶ were dilated from center đđ.

a. What is the scale factor đđ? Show your work.

|đ¶đ¶đ©đ©âČ| = đđ|đ¶đ¶đ©đ©| đđ = đđ(đđ) đđ = đđ

b. If |đđđ¶đ¶| â 2.2, what is |đđđ¶đ¶âČ|?

|đ¶đ¶đȘđȘâČ| = đđ|đ¶đ¶đȘđȘ|

|đ¶đ¶đȘđȘâČ| â đđ(đđ.đđ)

|đ¶đ¶đȘđȘâČ| â đđđđ

c. How does the perimeter of âł đđđ”đ”đ¶đ¶ compare to the perimeter of âł đđđ”đ”âČđ¶đ¶âČ?

Perimeter âłđ¶đ¶đ©đ©đȘđȘ â đđ + đđ + đđ.đđ Perimeter âł đ¶đ¶đ©đ©âČđȘđȘâČ â đđ + đđđđ + đđđđ

Perimeter âłđ¶đ¶đ©đ©đȘđȘ â đđ.đđ Perimeter âł đ¶đ¶đ©đ©âČđȘđȘâČ â đđđđ

d. Was the perimeter of âł đđđ”đ”âČđ¶đ¶âČ equal to the perimeter of âł đđđ”đ”đ¶đ¶ multiplied by scale factor đđ? Explain.

Yes. The perimeter of âł đ¶đ¶đ©đ©âČđȘđȘâČ was five times the perimeter of âł đ¶đ¶đ©đ©đȘđȘ, which makes sense because the dilation increased the length of each segment by a scale factor of đđ. That means that each side of âłđ¶đ¶đ©đ©âČđȘđȘâČ was five times as long as each side of âłđ¶đ¶đ©đ©đȘđȘ.

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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 đ”đ” on the đ„đ„-axis on the same vertical line as point đŽđŽ. Then we used what we learned in the last lesson. I should review my classwork.

Now I know which vertical line đŽđŽâČ will fall on; it is the same as where đ”đ”âČ is.

G8-M3-Lesson 5: First Consequences of FTS

1. Dilate point đŽđŽ, located at (2, 4) from center đđ, by a scale factor of đđ = 72. What is the precise location of point đŽđŽâČ?

|đ¶đ¶đ©đ©âČ| = đđ|đ¶đ¶đ©đ©|

|đ¶đ¶đ©đ©âČ| =đđđđ

(đđ)

|đ¶đ¶đ©đ©âČ| = đđ

|đšđšâČđ©đ©âČ| = đđ|đšđšđ©đ©|

|đšđšâČđ©đ©âČ| =đđđđ

(đđ)

|đšđšâČđ©đ©âČ| =đđđđđđ

|đšđšâČđ©đ©âČ| = đđđđ

I know that segment đŽđŽâČđ”đ”âČ is a dilation of segment đŽđŽđ”đ” by the same scale factor đđ = 7

2. Since đŽđŽđ”đ”ïżœïżœïżœïżœ is

contained within a vertical line, I can easily determine its length is 4 units.

Therefore, đšđšâČ is located at (đđ,đđđđ).

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Lesson 5: First Consequences of FTS

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2. Dilate point đŽđŽ, located at (7, 5) from center đđ, by a scale factor of đđ = 37. Then, dilate point đ”đ”, located at

(7, 3) from center đđ, by a scale factor of đđ = 37. What are the coordinates of đŽđŽâČ and đ”đ”âČ? Explain.

The đđ-coordinate of point đšđšâČ is the same as the length of đšđšâČđȘđȘâČïżœïżœïżœïżœïżœïżœ. Since |đšđšâČđȘđȘâČ| = đđ|đšđšđȘđȘ|, then |đšđšâČđȘđȘâČ| = đđ

đđâ đđ = đđđđ

đđ. The location of point đšđšâČ is ïżœđđ, đđđđ

đđïżœ, or approximately (đđ,đđ.đđ). The đđ-coordinate of

point đ©đ©âČ is the same as the length of đ©đ©âČđȘđȘâČïżœïżœïżœïżœïżœïżœ. Since |đ©đ©âČđȘđȘâČ| = đđ|đ©đ©đȘđȘ|, then |đ©đ©âČđȘđȘâČ| = đđđđâ đđ = đđ

đđ. The location

of point đ©đ©âČ is ïżœđđ, đđđđïżœ, or approximately (đđ,đđ.đđ).

This is just like the last problem, but now I have to find coordinates for two points. Iâll need to mark a point đ¶đ¶ on the đ„đ„-axis on the same vertical line as points đŽđŽ and đ”đ”.

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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 đŽđŽđŽđŽđŽđŽ is shown on the coordinate plane below. The triangle is dilated from the origin by scale factor đđ = 2. Identify the coordinates of the dilated triangle đŽđŽâČđŽđŽâČđŽđŽâČ.

đšđš(âđđ,âđđ) â đšđšâČïżœđđ â (âđđ),đđ â (âđđ)ïżœ = đšđšâČ(âđđ,âđđ) đ©đ©(âđđ,đđ) â đ©đ©âČ(đđ â (âđđ),đđ â đđ) = đ©đ©âČ(âđđ,đđ) đȘđȘ(đđ,đđ) â đȘđȘâČ(đđ â đđ,đđ â đđ) = đȘđȘâČ(đđ,đđ)

The coordinates of the dilated triangle will be đšđšâČ(âđđ,âđđ), đ©đ©âČ(âđđ,đđ), đȘđȘâČ(đđ,đđ).

2. The figure đŽđŽđŽđŽđŽđŽđŽđŽ has coordinates đŽđŽ(3, 1), đŽđŽ(12, 9), đŽđŽ(â9, 3), and đŽđŽ(â12,â3). The figure is dilated from the origin by a scale factor đđ = 23. Identify the coordinates of the dilated figure đŽđŽâČđŽđŽâČđŽđŽâČđŽđŽâČ.

đšđš(đđ,đđ) â đšđšâČ ïżœđđđđâ đđ,

đđđđâ đđïżœ = đšđšâČ ïżœđđ,

đđđđïżœ

đ©đ©(đđđđ,đđ) â đ©đ©âČ ïżœđđđđâ đđđđ,

đđđđâ đđïżœ = đ©đ©âČ(đđ,đđ)

đȘđȘ(âđđ,đđ) â đȘđȘâČ ïżœđđđđâ (âđđ),

đđđđâ đđïżœ = đȘđȘâČ(âđđ,đđ)

đ«đ«(âđđđđ,âđđ) â đ«đ«âČ ïżœđđđđâ (âđđđđ),

đđđđâ (âđđ)ïżœ = đ«đ«âČ(âđđ,âđđ)

The coordinates of the dilated figure are đšđšâČ ïżœđđ, đđđđïżœ, đ©đ©âČ (đđ,đđ), đȘđȘâČ(âđđ, đđ), and đ«đ«âČ(âđđ,âđđ).

The work we did in class led us to the conclusion that when given a point đŽđŽ(đ„đ„,đŠđŠ), we can find the coordinates of đŽđŽâČ using the scale factor: đŽđŽâČ(đđđ„đ„, đđđŠđŠ). 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 đđ by scale factor đđ 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 đđ with scale factor đđ so that đđâČ = đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) and đđâČ = đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ). Show that ray đđđđ maps to ray đđâČđđâČ (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 đđ be a point on line đđđđ that is not between points đđ and đđ.)

a. đđ is a point on đđđđ.ïżœïżœïżœïżœïżœïżœâ By definition of dilation what is the name of đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ)?

By the definition of dilation, we know that đŒđŒâČ = đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«(đŒđŒ).

b. By the definition of dilation we know that ïżœđ¶đ¶đ·đ·âČïżœ

|đ¶đ¶đ·đ·|= đđ. What other two ratios are also equal to đđ?

By the definition of dilation, we know that ïżœđ¶đ¶đžđžâČïżœ

|đ¶đ¶đžđž|= ïżœđ¶đ¶đŒđŒ

âČïżœ|đ¶đ¶đŒđŒ|

= đđ.

c. By FTS, what do we know about line đđđđ and line đđâČđđâČ?

By FTS, we know that line đ·đ·đžđž and line đ·đ·âČđžđžâČ are parallel.

d. What does FTS tell us about line đđđđ and line đđâČđđâČ?

By FTS, we know that line đžđžđŒđŒ and line đžđžâČđŒđŒâČ are parallel.

e. What conclusion can be drawn about the line that contains đđđđïżœïżœïżœïżœïżœâ and the line đđâČđđâČ?

The line that contains đ·đ·đžđžïżœïżœïżœïżœïżœïżœâ is parallel to đžđžâČđŒđŒâČïżœâïżœïżœïżœïżœïżœïżœâ because đžđžâČđŒđŒâČïżœâïżœïżœïżœïżœïżœïżœâ is contained in the line đ·đ·âČđžđžâČ.

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Lesson 7: Informal Proofs of Properties of Dilation

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f. Informally prove that đđâČđđâČïżœïżœïżœïżœïżœïżœïżœïżœâ is a dilation of đđđđ.ïżœïżœïżœïżœïżœïżœâ

Using the information from parts (a)â(e), we know that đŒđŒ is a point on đ·đ·đžđž.ïżœïżœïżœïżœïżœïżœïżœâ We also know that the line that contains đ·đ·đžđžïżœïżœïżœïżœïżœïżœâ is parallel to line đžđžâČđŒđŒâČ. But we already know that đ·đ·đžđžïżœâïżœïżœïżœâ is parallel to đ·đ·âČđžđžâČïżœâïżœïżœïżœïżœïżœïżœâ . Since there can only be one line that passes through đžđžâČ that is parallel to đ·đ·đžđžïżœïżœïżœïżœïżœïżœâ , then the line that contains đ·đ·âČđžđžâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ and line đžđžâČđŒđŒâČ must coincide. That places the dilation of point đŒđŒ, đŒđŒâČ, on đ·đ·âČđžđžâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ , 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 đŽđŽđŽđŽđŽđŽ that has been dilated from center đđ by scale factor đđ = 3. It is noted by đŽđŽâČđŽđŽâČđŽđŽâČ. We also have a triangle đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ, which is congruent to triangle đŽđŽâČđŽđŽâČđŽđŽâČ (i.e., âł đŽđŽâČđŽđŽâČđŽđŽâČ â âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ). Describe the sequence of a dilation, followed by a congruence (of one or more rigid motions), that would map triangle đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ onto triangle đŽđŽđŽđŽđŽđŽ.

First, we must dilate triangle đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ from center đđ by scale factor đđ = 1

3 to shrink it to the size of triangle đŽđŽđŽđŽđŽđŽ. I will note this triangle

as đŽđŽâČâČâČđŽđŽâČâČâČđŽđŽâČâČâČ. Once I have the triangle to the right size, I can translate the dilated triangle, đŽđŽâČâČâČđŽđŽâČâČâČđŽđŽâČâČâČ, 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 đšđšâČâČđ©đ©âČâČđȘđȘâČâČ from center đ¶đ¶ by scale factor đđ = đđđđ to shrink it to the size of

triangle đšđšđ©đ©đȘđȘ. Next, we must translate the dilated triangle, noted by đšđšâČâČâČđ©đ©âČâČâČđȘđȘâČâČâČ, one unit up and three units to the left. This sequence of the dilation followed by the translation would map triangle đšđšâČâČđ©đ©âČâČđȘđȘâČâČ onto triangle đšđšđ©đ©đȘđȘ.

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Lesson 8: Similarity

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2. Triangle đŽđŽđŽđŽđŽđŽ is similar to triangle đŽđŽâČđŽđŽâČđŽđŽâČ (i.e., âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČđŽđŽâČđŽđŽâČ). Prove the similarity by describing a sequence that would map triangle đŽđŽâČđŽđŽâČđŽđŽâČ onto triangle đŽđŽđŽđŽđŽđŽ.

The scale factor that would magnify triangle đšđšâČđ©đ©âČđȘđȘâČ to the size of triangle đšđšđ©đ©đȘđȘ is đđ = đđđđ.

Sample description:

The sequence that would prove the similarity of the triangles is a dilation from a center by a scale factor of đđ = đđ

đđ, followed by a

translation along vector đšđšâČđšđšïżœïżœïżœïżœïżœïżœïżœâ , and finally, a rotation about point đšđš.

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 đŽđŽâČđŽđŽâČđŽđŽâČ is the same size as triangle đŽđŽđŽđŽđŽđŽ, I can describe a congruence to map triangle đŽđŽâČđŽđŽâČđŽđŽâČ onto triangle đŽđŽđŽđŽđŽđŽ.

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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, âł đŽđŽđŽđŽđŽđŽ âŒ âł đŽđŽâČđŽđŽâČđŽđŽâČ and âł đŽđŽâČđŽđŽâČđŽđŽâČ ~ âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ. Is âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ? If so, describe the dilation followed by the congruence that demonstrates the similarity.

Yes, âł đšđšđšđšđšđš~âł đšđšâČâČđšđšâČâČđšđšâČâČ because similarity is transitive. Since đđ|đšđšđšđš| = |đšđšâČâČđšđšâČâČ|, |đšđšđšđš| = đđ, and |đšđšâČâČđšđšâČâČ| = đđ, then đđ â đđ = đđ. Therefore, đđ = đđ. Then, a dilation from the origin by scale factor đđ = đđ makes âłđšđšđšđšđšđš the same size as âł đšđšâČâČđšđšâČâČđšđšâČâČ. Translate the dilated image of âłđšđšđšđšđšđš, âł đšđšâČâČâČđšđšâČâČâČđšđšâČâČâČ, đđđđ units to the right and đđ units up to map đšđšâČâČâČ to đšđšâČâČ. Next, rotate the dilated image about point đšđšâČâČ, đđđđ degrees clockwise. Finally, reflect the rotated image across line đšđšâČâČđšđšâČâČ. The sequence of the dilation and the congruence map âł đšđšđšđšđšđš onto âłđšđšâČâČđšđšâČâČđšđšâČâČ, demonstrating the similarity.

First I need to dilate âł đŽđŽđŽđŽđŽđŽ to the same size as âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ. A dilation from the origin by đđ = 2 will make them the same size. Then I can describe a congruence to map the dilated image onto âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ.

I remember that when I have to translate an image, it is best to do it so that corresponding points, like đŽđŽ and đŽđŽâČâČ, 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, âł đšđšđšđšđšđš ~ âł đšđšâČđšđšâČđšđšâČ. They are similar because they have two pairs of corresponding angles that are equal in measure, namely, ïżœâ đšđšïżœ = ïżœâ đšđšâČïżœ = đđđđđđÂ°, and |â đšđš| = |â đšđšâČ| = đđđđÂ°.

2. Are the triangles shown below similar? Present an informal argument as to why they are or are not similar.

No, âł đšđšđšđšđšđš is not similar to âł đšđšâČđšđšâČđšđšâČ. By the given information, ïżœâ đšđšïżœ â ïżœâ đšđšâČïżœ, and ïżœâ đšđšïżœ â ïżœâ đšđšâČïżœ.

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 đ¶đ¶ and â đ¶đ¶âČ, 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 âł đšđšđšđšđšđš is similar to âł đšđšâČđšđšâČđšđšâČ. We can use the triangle sum theorem to find out that ïżœâ đšđšâČïżœ = đđđđÂ°, but we do not have any information about ïżœâ đšđšïżœ or ïżœâ đšđšïżœ. 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 đ”đ”âČ.

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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 âł đŽđŽđŽđŽđŽđŽ and âł đŽđŽâČđŽđŽâČđŽđŽâČ. Use this information to answer parts (a)â(b).

a. Based on the information given, is âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČđŽđŽâČđŽđŽâČ? Explain.

Yes, âł đšđšđšđšđšđš ~ âł đšđšâČđšđšâČđšđšâČ. 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.

đđ.đđđđđđ.đđđđ=

đđ.đđđđ.đđ

đđ.đđđđđđâŠ = đđ.đđđđđđâŠ

Since the values of these ratios are equal, approximately đđ.đđđđđđ, the triangles are similar.

b. Assume the length of side đŽđŽđŽđŽïżœïżœïżœïżœ is 4.03. What is the length of side đŽđŽâČđŽđŽâČïżœïżœïżœïżœïżœïżœ?

Let đđ represent the length of side đšđšâČđšđšâČïżœïżœïżœïżœïżœïżœ. đđ

đđ.đđđđ=đđ.đđđđ.đđđđ

We are looking for the value of đđ that makes the fractions equivalent. Therefore, đđ.đđđđđđ = đđđđ.đđđđđđ, and đđ = đđ.đđ. The length of side đšđšâČđšđšâČïżœïżœïżœïżœïżœïżœ is đđ.đđ.

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 đ„đ„ 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 âł đŽđŽđŽđŽđŽđŽ and âł đŽđŽâČđŽđŽđŽđŽâČ. Based on the information given, is âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČđŽđŽđŽđŽâČ? Explain.

Since both triangles have a common vertex, then |â đšđš| = |â đšđš|. This means that the measure of â đšđš in âł đšđšđšđšđšđš is equal to the measure of â đšđš in âł đšđšâČđšđšđšđšâČ. 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 đŽđŽđŽđŽ and đŽđŽâČđŽđŽâČ 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, âł đŹđŹđŹđŹđŹđŹ, and another formed by the person and his shadow, âłđ©đ©đ©đ©đŹđŹ. The triangles are similar if the height of the statue is measured at a đđđđÂ° angle with the ground and if the person standing forms a đđđđÂ° angle with the ground. We know that â đ©đ©đŹđŹđ©đ© is an angle common to both triangles. If |â đŹđŹđŹđŹđŹđŹ| = |â đ©đ©đ©đ©đŹđŹ| = đđđđÂ°, then âł đŹđŹđŹđŹđŹđŹ ~ âł đ©đ©đ©đ©đŹđŹ by the AA criterion.

2. Assume âł đžđžđžđžđžđž ~ âł đ”đ”đ”đ”đžđž. 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 đđ represent the height of the statue; then đđđđ =

đđđđđđ .

We are looking for the value of đđ that makes the fractions equivalent. Therefore, đđđđ = đđđđ, and đđ = đđđđ

đđ. The statue is about đđđđ feet tall.

If the height of the statue and I are measured at a 90Â° 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 đđ in the triangle below.

2. Determine the length of side đđ 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.

đđđđđđ + đđđđđđ = đđđđ đđđđđđ + đđđđđđ = đđđđ

đđđđđđ = đđđđ đđđđ = đđ

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.

đđđđ + đđđđ = đđ.đđđđ đđ + đđđđ = đđ.đđđđ

đđ â đđ + đđđđ = đđ.đđđđ â đđ đđđđ = đđ.đđđđ

đđ = đđ.đđ

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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 đđđđđđ + đđđđđđ = đđđđđđ is a true statement. The left side of the equation is equal to đđ,đđđđđđ. The right side of the equation is equal to đđ,đđđđđđ. That means đđđđđđ + đđđđđđ = đđđđđđ 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 đđđđđđ + đđđđđđ = đđđđđđ is a true statement. The left side of the equation is equal to đđ,đđđđđđ. The right side of the equation is equal to đđ,đđđđđđ. That means đđđđđđ + đđđđđđ = đđđđđđ is not true, and the triangle shown is not a right triangle.

I remember that đđ is the longest side of the triangle.

I need to check to see if đđ2 + đđ2 = đđ2 is true. If it is true, then it is right triangle.

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