Dilations

After a dilation with a center of (0, 0), a point was mapped as (1, 4) → (4, y). A student determined y to be 16. Evaluate the student's answer. A. The student is correct. B. The student incorrectly calculated the scale factor to be 4. C. The student incorrectly multiplied by the scale factor instead of adding it. D. The student incorrectly added the scale factor instead of multiplying by it.

A. The student is correct.

In a dilation with a scale factor of 3/4, the image is __________ the preimage. A. smaller than and along the same ray as B. larger than and along the same ray as C. smaller than and along the opposite ray from D. larger than and along the opposite ray from

A. smaller than and along the same ray as

Which of the following shows a dilation from triangle ABC to triangle JKL with a scale factor of 0.7? Assume that the center of dilation is the origin.

B

What are the coordinates of the image of the point (2, -6) under a dilation with a center of (0, 0) and a scale factor of 1/2? A. (0, 0) B. (1, -3) C. (2, -6) D. (4, -12)

B. (1, -3)

Determine the value of y. DO,k(-7, y) → (14, -5) A. 1.25 B. 2.5 C. 5 D. 10

B. 2.5

After a dilation with a center of (0, 0), a point was mapped as (2, -5) → (3, y). A student determined y to be -2.5. Evaluate the student's answer. A. The student is correct. B. The student incorrectly calculated the scale factor as 0.5. C. The student incorrectly divided by the scale factor instead of multiplying by it. D. The student incorrectly added the scale factor instead of multiplying by it.

B. The student incorrectly calculated the scale factor as 0.5

Figure G (blue) is the pre-image. Figure G' (red) is the image after a dilation. What type of dilation has been performed? A. enlargement B. reduction C. The image is the same size as the pre-image. D. no dilation has been performed

B. reduction

If point A is located at (-15, -7) and JKL has vertices at J(-22.5, -10.5), K(19.5, -3.0), and L(-4.5, 15.0), find the scale factor of the dilation from ABC to JKL. A. 0.4 B. 0.7 C. 1.5 D. 2.0

C. 1.5

Under a dilation, a figure is reduced and its orientation is preserved. Choose the statement that best describes the dilation. A. The image is along the same ray as the preimage but is farther from the center of dilation. B. The image is along the opposite ray from the preimage but is farther from the center of dilation. C. The image is along the same ray as the preimage but is closer to the center of dilation. D. The image is along the opposite ray from the preimage but is closer to the center of dilation.

C. The image is along the same ray as the preimage but is closer to the center of dilation.

After a dilation with a center of (0, 0), a point was mapped as (4, -6) → (12, y). A student determined y to be -2. Evaluate the student's answer. A. The student is correct. B. The student incorrectly calculated the scale factor to be -2. C. The student incorrectly divided by the scale factor instead of multiplying by it. D. The student incorrectly added the scale factor instead of multiplying by it.

C. The student incorrectly divided by the scale factor instead of multiplying by it.

A dilation with a scale factor, k, greater than 0 and less than 1 is a(n) __________. A. enlargement B. exact copy C. reduction D. translation

C. reduction

What are the coordinates of the image of the point (2, -4) after a dilation with a center of (0, 0) and scale factor of 4? A. (0, 0) B. (1/2, -1) C. (2, -4) D. (8, -16)

D. (8, -16)

What are the coordinates of the image of the point (4, 2) after a dilation with a center of (0, 0) and a scale factor of 2? A. (0, 0) B. (2, 1) C. (4, 2) D. (8, 4)

D. (8, 4)

Under a dilation centered at the origin, the image is congruent to the preimage. What is the scale factor? A. -1 or 0 B. 0 or 1 C. only 1 D. -1 or 1

D. -1 or 1

The coordinates of the vertices of triangle ABC are A(9, -10), B(-15, 0), and C(15, 10). A dilation centered at (0, 0) is performed on triangle ABC to produce triangle A'B'C'. The coordinates of the vertices of triangle A'B'C' are A'(13.5, -15), B'(-22.5, 0), and C'(22.5, 15). Find the scale factor. A. 0.5 B. 0.67 C. 1 D. 1.5

D. 1.5

Determine the value of x. DO,k(x, 9) → (-10, -6) A. 5 B. 7 C. 12 D. 15

D. 15

The scale factor, k, is 2. How does the size of the image relate to the pre-image? A. The image is 1/4 as large as the pre-image. B. The image is 1/2 as large as the pre-image. C. The image is the same size as the pre-image. D. The image is two times the size as the pre-image.

D. The image is two times the size as the pre-image.

Figure A is the preimage. Which figure is the image of figure A after a dilation with a scale factor of 3 and a center of (0, 0)? A. figure R B. figure S C. figure T D. figure U

D. figure U

scale factor of a dilation

In a dilation, the constant ratio between the distance from the center of dilation and a point on the image and the distance from the center of dilation and the corresponding point on the preimage.

center of dilation

In a dilation, the fixed point that is collinear with each point on the preimage and the corresponding point on the image.

dilation

In a plane, a transformation in which each point on the image lies on the same line as the corresponding point on the preimage and a fixed point called the center of dilation; the ratio between the distance from the center of dilation to a point on the image and the distance from the center of dilation to the corresponding point on the preimage remains constant.

Analyze the diagram below and complete the instructions that follow. A. k = 1/7 b. k = 5/7 c. k = 7/5 d. k = 5

NOT C

Under a dilation, a figure is enlarged and its orientation is preserved. Choose the statement that best describes the dilation. A. The image is along the same ray as the preimage but is farther from the center of dilation. B. The image is along the opposite ray from the preimage but is farther from the center of dilation. C. The image is along the same ray as the preimage but is closer to the center of dilation. D. The image is along the opposite ray from the preimage but is closer to the center of dilation.

NOT C

Section 4

PROFESSOR: Welcome back to Dilation. The objective for this section is to use an algebraic rule to describe or perform a dilation in the coordinate plane. So what if we have a negative scale factor? Well, when a scale factor is negative, the dilation of each point occurs along the opposite ray, and results in a point reflection across the center of dilation of the corresponding positive dilation. 00:00:26 So what does this mean? Let's take a look at an example. Draw the image of segment AB under a dilation with respect to the origin that has a skill factor of negative 2. So here's our center of dilation. Let's take a look at just the endpoints of AB. A is currently at 2, 4. And we can use the same rule, multiplying by negative 2 each 00:00:52 of those coordinates to get that A prime is at 2 times negative 2, which is negative 4, 4 times negative 2, which is negative 8. B is currently at 4, 1. So B prime is at negative 8, negative 2. Graphing this on our coordinate axes, negative 4, negative 8 is down here. So this is A prime. 00:01:25 And B prime is at negative 8, negative 2, which is here. That's B prime. So here's segment A prime, B prime. Notice that not only is it a point reflection of segment AB, but it is also larger than segment AB. And we can recognize negative scale factor dilation when it is a point reflection, or in other words, the dilation that's occurred along that opposite ray from how we would 00:02:02 normally do a dilation with a positive scale factor, and you can see that the center of dilation is in between the image and the preimage. And this doesn't occur with a positive scale factor. Graph triangle ABC with vertices A at 1, 2, B at 7, 2, C at 5, 5. And graph its image under a dilation with center at the origin and scale factor r equals negative 1. 00:02:43 So here's our center of dilation. Our scale factor is negative 1, so we can just think about multiplying these coordinates by negative 1. So A prime will be at negative 1, negative 2. B prime will be at negative 7, negative 2. And C prime we'll be at negative 5, negative 5. So graphing A prime, B prime, C prime. Negative 1, negative 2. 00:03:15 Here is A prime. B prime, negative 7, negative 2. And C prime, negative 5, negative 5. Connecting our vertices, we can see that triangle A prime, B prime, C prime is the same size as triangle ABC. Notice that our scale factor is negative, but it's still a 1. So this time it's not an identity transformation, but 00:03:47 it did preserve the size of the figure. Graph figure ABCD with vertices A at 2, 3, B at 3, 1, C at 9, 3, and D at 6, 1. And graph its image under a dilation with center at the origin and scale factor of r equals negative 1/3. So here's our center. And we should map our points. A prime will be at 2 times negative 1/3, negative 2/3, 3 00:04:29 times negative 1/3 is negative 1. B prime is at negative 1, negative 1/3. C prime will be at negative 3, negative 1. And D prime will be at negative 2, negative 1/3. So let's graph these image points. A prime is at negative 2/3, negative 1, way down here. B prime is at negative 1, negative 1/3, here. C prime is at negative 3, negative 1. 00:05:13 And D prime is at negative 2, negative 1/3. So connecting those vertices, I can see clearly that A prime, B prime, C prime, C prime is much smaller than ABCD. So this has resulted in a point reflection as well as a reduction. So let's review the differences between those both positive and negative scale factors. 00:05:41 Starting with the positive scale factors on the right- hand side, if r is greater than 0, we're going to be doing the dilation along the ray that extends from the center of dilation through the preimage. And if that scale factor is greater than 0 but less than 1, then it's going to be a reduction. If the scale factor is equal to 1 exactly, we get a congruent figure that's also an identity transformation, 00:06:17 meaning it's in the same position. And if the scale factor is greater than 1, then we have an enlargement. What about if the scale factor is negative? Well, then we are doing the dilation along the opposite ray. And another way to look at it is if the center is between the image and the preimage. 00:06:53 If the scale factor is greater than negative 1, in other words, it's between 0 and negative 1, then this is going to be a reduction. If the scale factor is equal to negative 1, we're still going to get congruent figures. It's not an identity transformation in this case. And if the scale factor is less than negative 1, meaning a large negative number, this is an enlargement. 00:07:16 Comparing these two, the difference between the negative scale factor and positive scale factor just makes a difference in which direction the ray of the dilation is going. And the scale factor of greater than negative 1 or less than 1, actually we can rewrite this as the absolute value of that scale factor is less than 1. If the absolute value of the scale factor is equal to 1, 00:07:45 then we're going to get congruent figures. And if the absolute value of the scale factor is greater than 1, thane we'll get an enlargement. So the red figure is the image of the blue figure after a dilation. Describe the scale factor of the dilation. So if we connect corresponding points on the image and preimage, I see that those lines are sort of converging 00:08:12 out here somewhere. This is going to be our center of dilation. So that center is not in between the preimage and image. So this is a positive scale factor. And I also see that the image is smaller than the preimage, meaning that it's reduced in size. So this is a reduction, meaning that it must also be 00:08:34 less than 1. And if we connect, on the second one, the corresponding points-- it's a little bit easier if we don't use vertices this time-- on the preimage and image, I see that that center of dilation where those lines are converging is in between the two figures. So this is going to be a negative scale factor. 00:09:09 And it's hard to see exactly, but it looks like the image is about the same size as the preimage, so this is probably about a scale factor of negative 1. But it's hard to say exactly without numbers. And last but not least, if I connect those corresponding points with lines, I see that they will be converging in a place that is not in between the two figures. So this is going to be a positive scale factor. 00:09:54 And the image is larger than the preimage, meaning it's grown in size. It's enlarged. So this is going to be a scale factor of that is greater than 1. Wrapping up for this section. Remember that a negative factor dilates each point along the opposite ray. 00:10:17 A scale factor absolute value less than 1 results in a reduction. If your scale factor has an absolute value greater than 1, it results in an enlargement. And if the absolute value of the scale factor is equal to 1, that results in a congruent image and preimage. Nice job working with dilations. See you next time.

Section 1

PROFESSOR: Welcome back. Think about it. What do you see happening here? You've probably heard about your pupils dilating in your eyes. And you've certainly seen someone blowing up a balloon. Well, these are both every day examples of the transformation that in mathematics we call dilations. 00:00:18 The objective for this section is to use an algebraic rule to describe or perform a dilation in the coordinate plane. What is a dilation? Well, in a plane it's a transformation in which each point on the image lies on the same line of the corresponding point on the preimage, in a fixed point called the center of dilation. What does that mean? 00:00:37 Let's take a look at our diagram here. This point out here is the center of dilation for the dilation of segment AB to create image segment a prime, b prime. So we do use the convention of adding those apostrophes to indicate the image as opposed to the preimage. And we can see that if we extend a ray or a line from the center of dilation through a point on the preimage, the 00:01:09 corresponding point on the image is also on that same line. Now this must be true in a dilation, and this next point must also be true. The ratio between the distance from the center of dilation to a point on the image and the distance from the center of dilation to the corresponding point on the preimage remains constant. 00:01:28 And this ratio is known as the scale factor. Now what does that mean? Let's take a look again at this example. We're talking about a ratio, the distance from the center of dilation to a point on the image, say, a prime. That distance is 8. We're taking a ratio of that to the distance from the center of dilation to its corresponding point. 00:01:51 Its corresponding point is a. So that distance is 4. So this ratio is equal to 2. Now, it should be true that that ratio remains constant. What does that mean? Well, if we do it again for a different point on the image, say b prime, this distance is 6. The ratio of that to the distance to its corresponding 00:02:13 point, which is b, is 3. So that ratio is also equal to 2. So our ratio is remaining constant. And that ratio is known as our scale factor. And we typically use, at least in this lesson we will use r to represent scale factor. So r equals 2. You may also see in other places letters such as k. 00:02:37 So the scale factor determines the type of dilation. And there are two main types of dilation. We have an enlargement, talking just about the size getting bigger. Or a reduction, where the size is getting smaller. So the size of the image in an enlargement is larger than the size of the preimage, and vice versa for the reduction. We can do dilations in the coordinate plane. 00:03:04 When we do this, we almost always have the center of dilation at the origin. So you can see if I extend a line or a ray from the center of dilation through any point on the preimage, its corresponding point on the image is also on that same line. And it is also true that that ratio remains constant, even on the coordinate plane. 00:03:31 And we can use coordinates to help us determine that ratio. A dilation is defined by two things, the center of dilation and the scale factor. So we have to use some notation to convey that information. We use a capital D for dilation. And we don't have to indicate the center dilation. If it's not indicated, we can safely assume that it is the 00:03:55 origin, because it almost always is. But you can indicate the center of dilation, capital O usually for origin, comma the scale factor maps x, y to rx, ry. It's important to note here that this is only true when the center of dilation is at the origin. But when this happens, it's really nice, because there is just one rule for all types of dilations that occur centered 00:04:30 at the origin. We can just multiply the scale factor by each one of the coordinates to find the coordinates of the image. "Given that the center of dilation is the origin, determine the scale factor." There's two ways we can do this. We can use the distances from the center of dilation to a point on the image, as opposed to a point on the preimage, 00:04:58 and find the ratio that way. Or we can use the mapping of x, y to rx, ry to find r. So let's try that way. C is at negative 15, 0. And it maps to C prime, which is at negative 10, 0. Because it's centered at the origin, I know that this is x, y mapping to rx, ry. So negative 10 must be equal to rx. 00:05:42 But I also know x. x is negative 15. So negative 10 equals r times negative 15. Divide both sides by negative 15. r equals negative 10 over negative 15, which is 2/3. So our scale factor is 2/3. This is a reduction. It's getting smaller. 00:06:13 It is always true that when we have a positive scale factor and it's less than 1, it's a reduction. Let's look at another example. "Given that the center of dilation is the origin, determine the scale factor." Well, here's our center of dilation. Here's our preimage. Here's our image. 00:06:32 Well, it's really in the same location. A dilation has occurred, but the image and the preimage are in the same location. They're the same size. It hasn't changed size. It's not an enlargement. It's not a reduction. This is what's called an identity transformation, or an 00:06:48 identity dilation. And that means that the original, the preimage figure, and the image figure really have the same size, same orientation, the same location. It's maintained its identity, so to speak. And this always occurs when we have a positive scale factor of 1. "Graph triangle ABC with vertices negative 6, negative 00:07:17 2, B at negative 2, 2, and C at 4, 0. And graph A prime, B prime, C prime under a dilation with center at the origin and A prime at negative 9, negative 3." So here's A prime out here. Knowing A prime, we can use our rule for mapping and find the scale factor, and use that to determine the mappings of the other two points in the image. So I know, because I'm centered at the origin, that 00:07:57 x, y maps to rx, ry. And I have an example of that with A and A prime. So A is negative 6, negative 2, which maps to negative 9, negative 3. So rx equals negative 9. But x is negative 6, so r times negative 5 equals negative 9. Divide both sides by negative 6 and I get the r equals 00:08:32 negative 9 over negative 9, or r equals 3/2. So I can use this scale factor to find the mappings of the other points. B prime will be at negative 2 times 3/2 is negative 3. And 2 times 3/2 is 3. So B prime is at negative 3, 3. Here's B prime. And C prime is at 4 times 3/2, which is 12/2, 00:09:16 which is 6 and 0. So here is C prime. So I have my other two points in my image figure A prime, B prime, C prime. In this case, my positive scale factor was greater than 1. And it was an enlargement. So it is always true, actually, that when my 00:09:42 positive scale factor is greater than 1, it results in an enlargement. Wrapping up for this section, remember a positive scale factor less than 1 results in a reduction. A positive scale factor equal to 1 results in an identity dilation. And a positive scale factor greater than 1 results in an enlargement. 00:10:00 Nice job working with those dilations. Stay tuned for more.

Sarah has built a scale model of a bridge. The real bridge is 56 ft across and 21 ft wide. Sarah's model is 8 in. across and 2.5 in. wide. Is Sarah's model a true model of the bridge? Explain.

Responses may vary but should include some or all of the following information: A dilation is a transformation where the object changes size by a scale factor. The preimage and image are similar: they have the same shape, same parallel lines, same angles, etc., and only differ in size by a scale factor. The scale factor, k, must be the same for each line or portion of the object. Measured across, Sarah's model has a scale factor of 8 in.:56 ft, or 1:84. Measured along the width, Sarah's model has a scale factor of 2.5 in.:21 ft, or 1: 100.8. This is not a true dilation because the scale factor of the length and the scale factor of the width are not the same.

Which of the following scale factors results in an enlargement of an image along an opposite ray? a. -2 b. -1/2 c. 1-2 d. 2

a. -2

AB has a length of 5 cm. The line segment is dilated to produce A'B', which has a length of 2 cm. Find the scale factor of the dilation. a. k = 2/5 b. k = 1/2 c. k = 5/2 d. k = 2

a. k = 2/5

Calculate the scale factor of the dilation around the center, C, origin. The pre-image is blue and the image is red. a. k = 1/5 b. k = 1/4 c. k = 1/2 d. k = 2

c. k = 1/2