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Module 3 Lessons 1â14 - Eagle Ridge Middle SchoolÂ Â· 2015-16 Lesson 2 : 4Properties of Dilations 8âą3 2. Use a compass to dilate the figure đŽđŽđŽđŽđŽđŽ from center

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Lessons 1â14

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• 2015-16

Lesson 1: What Lies Behind âSame Shapeâ?

8âą3

G8-M3-Lesson 1: What Lies Behind âSame Shapeâ?

1. Let there be a dilation from center đđ. Then, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ, and đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ. Examine thedrawing below. What can you determine about the scale factor of the dilation?

The dilation must have a scale factor larger than đđ, đđ > đđ, since the dilated points are farther from the center than the original points.

2. Let there be a dilation from center đđ with a scale factor đđ = 2. Then, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ, andđ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) = đđâČ. |đđđđ| = 1.7 cm, and |đđđđ| = 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 |đđđđâČ|.|đ¶đ¶đ¶đ¶âČ| = đđ|đ¶đ¶đ¶đ¶|; therefore, |đ¶đ¶đ¶đ¶âČ| = đđ â (đđ.đđ) = đđ.đđ and |đ¶đ¶đ¶đ¶âČ| = đđ.đđ đđđđ.

b. Use the definition of dilation to determine |đđđđâČ|.|đ¶đ¶đ¶đ¶âČ| = đđ|đ¶đ¶đ¶đ¶|; therefore, |đ¶đ¶đ¶đ¶âČ| = đđ â (đđ.đđ) = đđ.đđ and |đ¶đ¶đ¶đ¶âČ| = đđ.đđ đđđđ.

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, |đđđđâČ| is said, âThe length of segment đđđđ 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 âSame Shapeâ?

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3. Let there be a dilation from center đđ with a scale factor đđ. Then, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đ”đ”) = đ”đ”âČ, đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đ¶đ¶) = đ¶đ¶âČ, and |đđđ”đ”| = 10.8 cm, |đđđ¶đ¶| = 5 cm, and |đđđ”đ”âČ| = 2.7 cm, as shown. Use the drawing below to answer parts (a)â(c).

a. Using the definition of dilation with |đđđ”đ”| and |đđđ”đ”âČ|, determine the scale factor of the dilation.

|đ¶đ¶đ¶đ¶âČ| = đđ|đ¶đ¶đ¶đ¶|, which means đđ.đđ = đđ â (đđđđ.đđ), then

đđ.đđđđđđ.đđ

= đđ đđđđ

= đđ

b. Use the definition of dilation to determine |đđđ¶đ¶âČ|.

Since the scale factor, đđ, is đđđđ, then |đ¶đ¶đ¶đ¶âČ| = đđ

đđ|đ¶đ¶đ¶đ¶|;

therefore, |đ¶đ¶đ¶đ¶âČ| = đđđđâ đđ = đđ.đđđđ, and |đ¶đ¶đ¶đ¶âČ| =

đđ.đđđđ đđđđ.

Since |đđđ”đ”âČ| = đđ|đđđ”đ”|, then by the multiplication property of

equality, ïżœđđđ”đ”âČïżœ

|đđđ”đ”|= đđ.

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Lesson 2: Properties of Dilations

8âą3

G8-M3-Lesson 2: Properties of Dilations

1. Given center đđ and quadrilateral đŽđŽđŽđŽđŽđŽđŽđŽ, use a ruler to dilate the figure from center đđ by a scale factor of đđ = 1

4. Label the dilated quadrilateral đŽđŽâČđŽđŽâČđŽđŽâČđŽđŽâČ.

The figure in red below shows the dilated image of đšđšđšđšđšđšđšđš. All measurements are in centimeters.

|đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš| |đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš| |đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš| |đ¶đ¶đšđšâČ| = đđ|đ¶đ¶đšđš|

|đ¶đ¶đšđšâČ| = đđđđ

(đđđđ.đđ) |đ¶đ¶đšđšâČ| = đđđđ

(đđ.đđ) |đ¶đ¶đšđšâČ| = đđđđ

(đđ.đđ) |đ¶đ¶đšđšâČ| = đđđđ

(đđ.đđ)

|đ¶đ¶đšđšâČ| = đđ.đđ |đ¶đ¶đšđšâČ| = đđ.đđ |đ¶đ¶đšđšâČ| = đđ.đđ |đ¶đ¶đšđšâČ| = đđ.đđ

I need to draw the rays from the center đđ to each point on the figure and then measure the length from đđ 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 đŽđŽđŽđŽđŽđŽ from center đđ, with scale factor đđ = 3.

The figure in red below shows the dilated image of đšđšđšđšđšđš.

I need to draw the rays like before, but this time I can use a compass to measure the distance from the center đđ 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 đŽđŽđŽđŽđŽđŽđŽđŽ from center đđ, with scale factor đđ = 2. a. Dilate the same figure, đŽđŽđŽđŽđŽđŽđŽđŽ, from a new center, đđâČ, with scale factor đđ = 2. Use double primes

(đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ) to distinguish this image from the original.

The figure in blue, đšđšâČđšđšâČđšđšâČđšđšâČ, shows the dilation of đšđšđšđšđšđšđšđš from center đ¶đ¶, with scale factor đđ = đđ. The figure in red, đšđšâČâČđšđšâČâČđšđšâČâČđšđšâČâČ, shows the dilation of đšđšđšđšđšđšđšđš from center đ¶đ¶âČ with scale factor đđ = đđ.

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 đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ to đŽđŽâČđŽđŽâČđŽđŽâČđŽđŽâČ?

A translation along vector đšđšâČâČđšđšâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ (or any vector that connects a point of đšđšâČâČđšđšâČâČđšđšâČâČđšđšâČâČ and its corresponding point of đšđšâČđšđšâČđšđšâČđšđšâČ) would map the figure đšđšâČâČđšđšâČâČđšđšâČâČđšđšâČâČ to đšđšâČđšđšâČđšđšâČđšđšâČ.

The image below (with rays removed for clarity) shows the vector đšđšâČâČđšđšâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ .

4. A line segment đŽđŽđŽđŽ undergoes a dilation. Based on todayâs lesson, what is the image of the segment?

The segment dilates as a segment.

5. â đŽđŽđđđŽđŽ measures 24Â°. After a dilation, what is the measure of â đŽđŽâČđđđŽđŽâČ? How do you know?

The measure of â đšđšâČđ¶đ¶đšđšâČ is đđđđÂ°. Dilations preserve angle measure, so â đšđšâČđ¶đ¶đšđšâČ remains the same measure as â đšđšđ¶đ¶đšđš.

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Lesson 3: Examples of Dilations

8âą3

G8-M3-Lesson 3: Examples of Dilations

1. Dilate the figure from center đđ by a scale factor of đđ = 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”, đžđž, 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 đŽđŽđ”đ”đŽđŽ was dilated from center đđ by a scale factor of đđ = 8. What scale factor would shrink the dilated figure back to the original size?

A scale factor of đđ = đđđđ would bring the dilated figure

back to its original size.

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

A scale factor of đđ = đđđđ 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 đđ â 1

đđ= 1.

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Lesson 4: Fundamental Theorem of Similarity (FTS)

8âą3

G8-M3-Lesson 4: Fundamental Theorem of Similarity (FTS)

1. In the diagram below, points đđ, đđ, and đŽđŽ have been dilated from center đđ by a scale factor of đđ = 1

5 on a piece of lined paper.

a. What is the relationship between segments đđđđ and đđâČđđâČ? How do you know?

đ·đ·đ·đ·ïżœïżœïżœïżœ and đ·đ·âČđ·đ·âČïżœïżœïżœïżœïżœïżœ are parallel because they follow the lines on the lined paper.

b. What is the relationship between segments đđđŽđŽ and đđâČđŽđŽâČ? How do you know?

đ·đ·đ·đ·ïżœïżœïżœïżœ and đ·đ·âČđ·đ·âČïżœïżœïżœïżœïżœïżœ 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·âČđ·đ·âČ are equal in measure because they are corresponding angles created by parallel lines đ·đ·đ·đ· and đ·đ·âČđ·đ·âČ.

d. What is the relationship between the lengths of segments đŽđŽđđ and đŽđŽâČđđâČ? How do you know?

The length of segment đ·đ·âČđ·đ·âČ will be đđđđ 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·âČđ·đ·âČ| = đđ|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 đđ.

a. What is the scale factor đđ? Show your work.

= đđ

đđ = đđ

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

= đđ

đđ = đđ

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

d. Which angles are equal in measure? How do you know?

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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 đđ.

a. What is the scale factor đđ? Show your work.

b. If |đđđ¶đ¶| â 2.2, what is |đđđ¶đ¶âČ|?

|đ¶đ¶đȘđȘâČ| = đđ|đ¶đ¶đȘđȘ|

|đ¶đ¶đȘđȘâČ| â đđ(đđ.đđ)

|đ¶đ¶đȘđȘâČ| â đđđđ

c. How does the perimeter of âł đđđ”đ”đ¶đ¶ compare to the perimeter of âł đđđ”đ”âČđ¶đ¶âČ?

d. Was the perimeter of âł đđđ”đ”âČđ¶đ¶âČ equal to the perimeter of âł đđđ”đ”đ¶đ¶ multiplied by scale factor đđ? Explain.

Yes. The perimeter of âł đ¶đ¶đ©đ©âČđȘđȘâČ was five times the perimeter of âł đ¶đ¶đ©đ©đȘđȘ, which makes sense because the dilation increased the length of each segment by a scale factor of đđ. That means that each side of âłđ¶đ¶đ©đ©âČđȘđȘâČ was five times as long as each side of âłđ¶đ¶đ©đ©đȘđȘ.

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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 đŽđŽ. Then we used what we learned in the last lesson. I should review my classwork.

Now I know which vertical line đŽđŽâČ will fall on; it is the same as where đ”đ”âČ is.

G8-M3-Lesson 5: First Consequences of FTS

1. Dilate point đŽđŽ, located at (2, 4) from center đđ, by a scale factor of đđ = 72. What is the precise location of point đŽđŽâČ?

(đđ)

(đđ)

I know that segment đŽđŽâČđ”đ”âČ is a dilation of segment đŽđŽđ”đ” by the same scale factor đđ = 7

2. Since đŽđŽđ”đ”ïżœïżœïżœïżœ is

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

Therefore, đšđšâČ is located at (đđ,đđđđ).

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• 2015-16

Lesson 5: First Consequences of FTS

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2. Dilate point đŽđŽ, located at (7, 5) from center đđ, by a scale factor of đđ = 37. Then, dilate point đ”đ”, located at

(7, 3) from center đđ, by a scale factor of đđ = 37. What are the coordinates of đŽđŽâČ and đ”đ”âČ? Explain.

The đđ-coordinate of point đšđšâČ is the same as the length of đšđšâČđȘđȘâČïżœïżœïżœïżœïżœïżœ. Since |đšđšâČđȘđȘâČ| = đđ|đšđšđȘđȘ|, then |đšđšâČđȘđȘâČ| = đđ

đđâ đđ = đđđđ

đđ. The location of point đšđšâČ is ïżœđđ, đđđđ

đđïżœ, or approximately (đđ,đđ.đđ). The đđ-coordinate of

đđ. The location

This is just like the last problem, but now I have to find coordinates for two points. Iâll need to mark a point đ¶đ¶ on the đ„đ„-axis on the same vertical line as points đŽđŽ and đ”đ”.

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• 2015-16

Lesson 6: Dilations on the Coordinate Plane

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G8-M3-Lesson 6: Dilations on the Coordinate Plane

1. Triangle đŽđŽđŽđŽđŽđŽ is shown on the coordinate plane below. The triangle is dilated from the origin by scale factor đđ = 2. Identify the coordinates of the dilated triangle đŽđŽâČđŽđŽâČđŽđŽâČ.

2. The figure đŽđŽđŽđŽđŽđŽđŽđŽ has coordinates đŽđŽ(3, 1), đŽđŽ(12, 9), đŽđŽ(â9, 3), and đŽđŽ(â12,â3). The figure is dilated from the origin by a scale factor đđ = 23. Identify the coordinates of the dilated figure đŽđŽâČđŽđŽâČđŽđŽâČđŽđŽâČ.

đšđš(đđ,đđ) â đšđšâČ ïżœđđđđâ đđ,

đđđđâ đđïżœ = đšđšâČ ïżœđđ,

đđđđïżœ

đȘđȘ(âđđ,đđ) â đȘđȘâČ ïżœđđđđâ (âđđ),

đđđđâ đđïżœ = đȘđȘâČ(âđđ,đđ)

đ«đ«(âđđđđ,âđđ) â đ«đ«âČ ïżœđđđđâ (âđđđđ),

đđđđâ (âđđ)ïżœ = đ«đ«âČ(âđđ,âđđ)

The coordinates of the dilated figure are đšđšâČ ïżœđđ, đđđđïżœ, đ©đ©âČ (đđ,đđ), đȘđȘâČ(âđđ, đđ), and đ«đ«âČ(âđđ,âđđ).

The work we did in class led us to the conclusion that when given a point đŽđŽ(đ„đ„,đŠđŠ), we can find the coordinates of đŽđŽâČ using the scale factor: đŽđŽâČ(đđđ„đ„, đđđŠđŠ). This only works for dilations from the origin.

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• 2015-16

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 đđ by scale factor đđ 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 đđ with scale factor đđ so that đđâČ = đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ) and đđâČ = đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·đ·(đđ). Show that ray đđđđ maps to ray đđâČđđâČ (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 đđ be a point on line đđđđ that is not between points đđ and đđ.)

a. đđ is a point on đđđđ.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·(đđ)?

By the definition of dilation, we know that đŒđŒâČ = đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«đ«(đŒđŒ).

b. By the definition of dilation we know that ïżœđ¶đ¶đ·đ·âČïżœ

|đ¶đ¶đ·đ·|= đđ. What other two ratios are also equal to đđ?

By the definition of dilation, we know that ïżœđ¶đ¶đžđžâČïżœ

|đ¶đ¶đžđž|= ïżœđ¶đ¶đŒđŒ

âČïżœ|đ¶đ¶đŒđŒ|

= đđ.

c. By FTS, what do we know about line đđđđ and line đđâČđđâČ?

By FTS, we know that line đ·đ·đžđž and line đ·đ·âČđžđžâČ are parallel.

d. What does FTS tell us about line đđđđ and line đđâČđđâČ?

By FTS, we know that line đžđžđŒđŒ and line đžđžâČđŒđŒâČ are parallel.

e. What conclusion can be drawn about the line that contains đđđđïżœïżœïżœïżœïżœâ and the line đđâČđđâČ?

The line that contains đ·đ·đžđžïżœïżœïżœïżœïżœïżœâ is parallel to đžđžâČđŒđŒâČïżœâïżœïżœïżœïżœïżœïżœâ because đžđžâČđŒđŒâČïżœâïżœïżœïżœïżœïżœïżœâ is contained in the line đ·đ·âČđžđžâČ.

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Lesson 7: Informal Proofs of Properties of Dilation

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f. Informally prove that đđâČđđâČïżœïżœïżœïżœïżœïżœïżœïżœâ is a dilation of đđđđ.ïżœïżœïżœïżœïżœïżœâ

Using the information from parts (a)â(e), we know that đŒđŒ is a point on đ·đ·đžđž.ïżœïżœïżœïżœïżœïżœïżœâ We also know that the line that contains đ·đ·đžđžïżœïżœïżœïżœïżœïżœâ is parallel to line đžđžâČđŒđŒâČ. But we already know that đ·đ·đžđžïżœâïżœïżœïżœâ is parallel to đ·đ·âČđžđžâČïżœâïżœïżœïżœïżœïżœïżœâ . Since there can only be one line that passes through đžđžâČ that is parallel to đ·đ·đžđžïżœïżœïżœïżœïżœïżœâ , then the line that contains đ·đ·âČđžđžâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ and line đžđžâČđŒđŒâČ must coincide. That places the dilation of point đŒđŒ, đŒđŒâČ, on đ·đ·âČđžđžâČïżœïżœïżœïżœïżœïżœïżœïżœïżœâ , which proves that dilations map rays to rays.

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• 2015-16

Lesson 8: Similarity

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G8-M3-Lesson 8: Similarity

1. In the picture below, we have a triangle đŽđŽđŽđŽđŽđŽ that has been dilated from center đđ by scale factor đđ = 3. It is noted by đŽđŽâČđŽđŽâČđŽđŽâČ. We also have a triangle đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ, which is congruent to triangle đŽđŽâČđŽđŽâČđŽđŽâČ (i.e., âł đŽđŽâČđŽđŽâČđŽđŽâČ â âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ). Describe the sequence of a dilation, followed by a congruence (of one or more rigid motions), that would map triangle đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ onto triangle đŽđŽđŽđŽđŽđŽ.

First, we must dilate triangle đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ from center đđ by scale factor đđ = 1

3 to shrink it to the size of triangle đŽđŽđŽđŽđŽđŽ. I will note this triangle

as đŽđŽâČâČâČđŽđŽâČâČâČđŽđŽâČâČâČ. Once I have the triangle to the right size, I can translate the dilated triangle, đŽđŽâČâČâČđŽđŽâČâČâČđŽđŽâČâČâČ, 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 đšđšâČâČđ©đ©âČâČđȘđȘâČâČ from center đ¶đ¶ by scale factor đđ = đđđđ to shrink it to the size of

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Lesson 8: Similarity

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2. Triangle đŽđŽđŽđŽđŽđŽ is similar to triangle đŽđŽâČđŽđŽâČđŽđŽâČ (i.e., âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČđŽđŽâČđŽđŽâČ). Prove the similarity by describing a sequence that would map triangle đŽđŽâČđŽđŽâČđŽđŽâČ onto triangle đŽđŽđŽđŽđŽđŽ.

Sample description:

The sequence that would prove the similarity of the triangles is a dilation from a center by a scale factor of đđ = đđ

đđ, followed by a

translation along vector đšđšâČđšđšïżœïżœïżœïżœïżœïżœïżœâ , and finally, a rotation about point đšđš.

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 đŽđŽâČđŽđŽâČđŽđŽâČ is the same size as triangle đŽđŽđŽđŽđŽđŽ, I can describe a congruence to map triangle đŽđŽâČđŽđŽâČđŽđŽâČ onto triangle đŽđŽđŽđŽđŽđŽ.

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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, âł đŽđŽđŽđŽđŽđŽ âŒ âł đŽđŽâČđŽđŽâČđŽđŽâČ and âł đŽđŽâČđŽđŽâČđŽđŽâČ ~ âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ. Is âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ? If so, describe the dilation followed by the congruence that demonstrates the similarity.

Yes, âł đšđšđšđšđšđš~âł đšđšâČâČđšđšâČâČđšđšâČâČ because similarity is transitive. Since đđ|đšđšđšđš| = |đšđšâČâČđšđšâČâČ|, |đšđšđšđš| = đđ, and |đšđšâČâČđšđšâČâČ| = đđ, then đđ â đđ = đđ. Therefore, đđ = đđ. Then, a dilation from the origin by scale factor đđ = đđ makes âłđšđšđšđšđšđš the same size as âł đšđšâČâČđšđšâČâČđšđšâČâČ. Translate the dilated image of âłđšđšđšđšđšđš, âł đšđšâČâČâČđšđšâČâČâČđšđšâČâČâČ, đđđđ units to the right and đđ units up to map đšđšâČâČâČ to đšđšâČâČ. Next, rotate the dilated image about point đšđšâČâČ, đđđđ degrees clockwise. Finally, reflect the rotated image across line đšđšâČâČđšđšâČâČ. The sequence of the dilation and the congruence map âł đšđšđšđšđšđš onto âłđšđšâČâČđšđšâČâČđšđšâČâČ, demonstrating the similarity.

First I need to dilate âł đŽđŽđŽđŽđŽđŽ to the same size as âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ. A dilation from the origin by đđ = 2 will make them the same size. Then I can describe a congruence to map the dilated image onto âł đŽđŽâČâČđŽđŽâČâČđŽđŽâČâČ.

I remember that when I have to translate an image, it is best to do it so that corresponding points, like đŽđŽ and đŽđŽâČâČ, 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, âł đšđšđšđšđšđš ~ âł đšđšâČđšđšâČđšđšâČ. They are similar because they have two pairs of corresponding angles that are equal in measure, namely, ïżœâ đšđšïżœ = ïżœâ đšđšâČïżœ = đđđđđđÂ°, and |â đšđš| = |â đšđšâČ| = đđđđÂ°.

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

No, âł đšđšđšđšđšđš is not similar to âł đšđšâČđšđšâČđšđšâČ. By the given information, ïżœâ đšđšïżœ â  ïżœâ đšđšâČïżœ, and ïżœâ đšđšïżœ â  ïżœâ đšđšâČïżœ.

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¶âČ, 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 âł đšđšđšđšđšđš is similar to âł đšđšâČđšđšâČđšđšâČ. We can use the triangle sum theorem to find out that ïżœâ đšđšâČïżœ = đđđđÂ°, but we do not have any information about ïżœâ đšđšïżœ or ïżœâ đšđšïżœ. 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”âČ.

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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 âł đŽđŽđŽđŽđŽđŽ and âł đŽđŽâČđŽđŽâČđŽđŽâČ. Use this information to answer parts (a)â(b).

a. Based on the information given, is âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČđŽđŽâČđŽđŽâČ? Explain.

Yes, âł đšđšđšđšđšđš ~ âł đšđšâČđšđšâČđšđšâČ. 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.

đđ.đđđđđđ.đđđđ=

đđ.đđđđ.đđ

đđ.đđđđđđâŠ = đđ.đđđđđđâŠ

Since the values of these ratios are equal, approximately đđ.đđđđđđ, the triangles are similar.

b. Assume the length of side đŽđŽđŽđŽïżœïżœïżœïżœ is 4.03. What is the length of side đŽđŽâČđŽđŽâČïżœïżœïżœïżœïżœïżœ?

Let đđ represent the length of side đšđšâČđšđšâČïżœïżœïżœïżœïżœïżœ. đđ

đđ.đđđđ=đđ.đđđđ.đđđđ

We are looking for the value of đđ that makes the fractions equivalent. Therefore, đđ.đđđđđđ = đđđđ.đđđđđđ, and đđ = đđ.đđ. The length of side đšđšâČđšđšâČïżœïżœïżœïżœïżœïżœ is đđ.đđ.

I donât 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.

24

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• 2015-16

Lesson 11: More About Similar Triangles

8âą3

2. In the diagram below, you have âł đŽđŽđŽđŽđŽđŽ and âł đŽđŽâČđŽđŽđŽđŽâČ. Based on the information given, is âł đŽđŽđŽđŽđŽđŽ ~ âł đŽđŽâČđŽđŽđŽđŽâČ? Explain.

Since both triangles have a common vertex, then |â đšđš| = |â đšđš|. This means that the measure of â đšđš in âł đšđšđšđšđšđš is equal to the measure of â đšđš in âł đšđšâČđšđšđšđšâČ. 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 đŽđŽđŽđŽ and đŽđŽâČđŽđŽâČ look like they might be parallel, I donât 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.

25

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Homework Helper A Story of Ratios

• 2015-16

Lesson 12: Modeling Using Similarity

8âą3

G8-M3-Lesson 12: Modeling Using Similarity

1. There is a statue of your schoolâs 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.

2. Assume âł đžđžđžđžđžđž ~ âł đ”đ”đ”đ”đžđž. 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 đđ represent the height of the statue; then đđđđ =

đđđđđđ .

We are looking for the value of đđ that makes the fractions equivalent. Therefore, đđđđ = đđđđ, and đđ = đđđđ

đđ. The statue is about đđđđ 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.

26

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Homework Helper A Story of Ratios

• 2015-16

Lesson 13: Proof of the Pythagorean Theorem

8âą3

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 đđ in the triangle below.

2. Determine the length of side đđ 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.

đđđđđđ + đđđđđđ = đđđđ đđđđđđ + đđđđđđ = đđđđ

đđđđđđ = đđđđ đđđđ = đđ

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.

đđđđ + đđđđ = đđ.đđđđ đđ + đđđđ = đđ.đđđđ

đđ â đđ + đđđđ = đđ.đđđđ â đđ đđđđ = đđ.đđđđ

đđ = đđ.đđ

27

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Homework Helper A Story of Ratios

• 2015-16

Lesson 14: The Converse of the Pythagorean Theorem

8âą3

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 đđđđđđ + đđđđđđ = đđđđđđ is a true statement. The left side of the equation is equal to đđ,đđđđđđ. The right side of the equation is equal to đđ,đđđđđđ. That means đđđđđđ + đđđđđđ = đđđđđđ 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 đđđđđđ + đđđđđđ = đđđđđđ is a true statement. The left side of the equation is equal to đđ,đđđđđđ. The right side of the equation is equal to đđ,đđđđđđ. That means đđđđđđ + đđđđđđ = đđđđđđ is not true, and the triangle shown is not a right triangle.

I remember that đđ is the longest side of the triangle.

I need to check to see if đđ2 + đđ2 = đđ2 is true. If it is true, then it is right triangle.

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