Question

The endpoints of line AB are A(9,4) and B(5,-4). The endpoints of the image after a dilation are A(6,3) and B(3,-3). Find the scale factor and explain each of your steps

Answer 1

Ok, let's graph this, and see if it can help you visualize it.

Answer 2

The points are plotted here:
https://www.desmos.com/calculator/mclsqhwnkc

Answer 3

What do you notice?

Answer 4

the first one is longer than the dialation one

Answer 5

Well, yeah. The red points are the endpoints of the translated segment.

Answer 6

I mean, how do you think we could find the scale of dilation?

Answer 7

yes i got that

Answer 8

The scale of dilation is, for example, take:

(x*3,y*3)

The dilation is a scale of 3

(x*?,y*?)

How do you think we can find the scale of dilation for this problem?

Answer 9

i dont know how to find the scale factor but i know it is less than 1

Answer 10

You're already on the right track.

Answer 11

I'm going to edit the coordinates on desmos a bit to help you visualize it.

Answer 12

ok

Answer 13

Ok, look on desmos

Answer 14

this is what i got graphing the coordinates

Answer 15

Look at the bottom

Answer 16

There are little sliders

Answer 17

Where it says n1 = 1
and n2 =1

Answer 18

If you play around with those, you can see the dilations visually.

Answer 19

Try and find A(6,3) and B(3,-3)

Answer 20

im confused what are we looking at

Answer 21

Do you see the little sliders that say n1 = 1 and n2 =1

Answer 22

On desmos

Answer 23

https://www.desmos.com/calculator/mclsqhwnkc

Answer 24

no

Answer 25

See it in the lower left corner?

Answer 26

I built a little dilation playground for you to help you understand :P

if you play around with those sliders, you can see the dilations real time

Answer 27

i dont have those on my screen

Answer 28

Oh, sorry there's a new link: https://www.desmos.com/calculator/qjmzzqfz46

Answer 29

Try now.

Answer 30

Click and drag the sliders to see the dilations.

Answer 31

ok i got them now so what am i doing with them

Answer 32

If you move the sliders around, you can visualize the dilations.

Answer 33

So, using these points do you think we can create A(6,3) and B(3,-3)?

Answer 34

im getting the red coordinates not the green

Answer 35

wait nevermind

Answer 36

Look, you have to move the sliders:

Answer 37

i got that but this is very confusing i cant get the sliders to the right coordinates

Answer 38

That's right.

Answer 39

We can't

Answer 40

Why?

Answer 41

ok ?

Answer 42

(By the way, sorry I'm dragging this out. I just want you to understand this thoroughly.)

Answer 43

(and I also want it to be fun :P)

Answer 44

Anyway, clearly we can't just multiply x and y by certain numbers to dilate certain figures.

Answer 45

Here's why.

Answer 46

When you multiply x and y by different variables, you assume that (0,0) (the origin) is the center of dilation.

Answer 47

If you can't multiply these variables, though, the only answer is, the origin is not our center point for dilation...

Answer 48

Which means, to figure out our scale of dilation, we can't just find these numbers, we're going to have to use ratios.

Answer 49

that makes no sense and i have to write down all my steps to find my scale factor and i cant do that doing it this way

Answer 50

Don't worry, you'll be able to.

Answer 51

All we have to do is find the center of the dilation, then we can find the scale easily.

Answer 52

Let me show you.

Answer 53

I'm sorry if this is confusing you :(

Answer 54

i just have no idea and i have alot going on in my head so its hard to focus on this but i have to do it

Answer 55

I'm so sorry. Look here: https://www.desmos.com/calculator/lttgxf6exs This is how I found the center of dilation. I just drew lines that went from the first A to the second A, and from the first B to the second B, and where they intersect must be the center of dilation.

Answer 56

So, the center is (-3,0)

Answer 57

Now, to find the scale.

Answer 58

ok

Answer 59

I haven't done this in a while, but I think it's just the slopes of the line...

Answer 60

i dont understand why we need those lines what are they what do they do to find the scale factor

Answer 61

So, I think it's (A*1/3) and (B*-1/2)

Answer 62

Let's check.

Answer 63

but what are the steps to get to that

Answer 64

Ugh... I'm so sorry stuck... I'm kinda lost. I'm trying, but I haven't done problems like this in a while. :( I think I may need some help.

Answer 65

I'm doing something wrong here.

Answer 66

@lochana Can you help me with this question?

Answer 67

He seems knowledgable about dilations.

Answer 68

I'm better with the basic matrix transformations.

Answer 69

I guess we'll just have to wait for @lochana because I want to make sure I'm not giving out incorrect answers. I just need to make sure I'm not giving out false answers.

Answer 70

That is the last thing I would want @stuck-help :(

Answer 71

thats fine i think im going to take a break for a little bit

Answer 72

Ok... once again, sorry...

Answer 73

it is a strange one:)

Answer 74

I know, right?

Answer 75

I found the center of dilation, but wasn't sure what to do from there... :P

Answer 76

https://www.desmos.com/calculator/lttgxf6exs

Answer 77

I think you can tackle it only by considering length of each line

Answer 78

OH YEAH

Answer 79

WHY DIDNT I THINK OF THAT?!?

Answer 80

[\scale = \frac{length of AB after slcaled}{length of AB before scaled}\]

Answer 81

I was like taking the slopes of the lines and saying, ok so its 1/3 and -1/2 but you can just use pythagorean theorem to find the lengths of the lines and then divide.

Answer 82

\[scale = \frac{length of AB after slcaled}{length of AB before scaled}\]

Answer 83

yeah. you can do in that way too. but you have to be wet little bit:)

Answer 84

So, to find the lengths we can just use pythagorean theorem.

\[AB = \sqrt{4^2+8^2}\]

\[A'B' = \sqrt{3^2+6^2}\]

Answer 85

\[AB length(before) = \sqrt{45}\]\[AB length(after) = \sqrt{80}\]\[scale = \frac{\sqrt{80}}{\sqrt{45}} = \frac{4}{3}\]

Answer 86

exactly

Answer 87

Thank you so much for the clarification ^_^

Answer 88

I was struggling so much with this... :P

Answer 89

you are welcome:)

Answer 90

ah that's fine. it is a good thing:)

Answer 91

I was on the right track, but there was a much simpler route.

Answer 92

So, @stuck-help, there's your answer.

Answer 93

@malcolmmcswain is this the start to it