6. Describe Dilations
Sign In
In a dilation, each coordinate of the points in the preimage is multiplied by the same number, the scale factor.
Scale factor: 25
Mistake our friend likely made: Mistaking △ ABC for image and △ A'B'C' for preimage
We are given a dilation with center at (0,0). Our friend says that the scale factor is 52. We want to find the correct scale factor, then find the mistake that our friend likely made.
Before we find the scale factor, let's include the coordinates of the vertices of the two triangles in the graph. We will need them later.
Now, in a dilation each coordinate of the points in the preimage is multiplied by the same number, the scale factor r. This gives us the coordinates of the points in the image. ccc Preimage & & Image [0.5em] (x,y)& ⇒ & ( rx, ry) Note that here the preimage is triangle ABC and the image is triangle A'B'C'. Let's now take a look at our points.
| Preimage | Image |
|---|---|
| A(0,5) | A'(0,2) |
| B(- 5,0) | B'(- 2,0) |
| C(5,- 5) | B'(2,- 2) |
We will focus on points A and A'. We need to consider what scale factor would get us from (0,5) to (0,2). Let's try 25. ( 25(0), 25(5)) = (0,2) As we can see, 0* 25=0 and 5* 25=2. This may be our scale factor. Let's check to see if our number works for the remaining points.
| Dilation With Scale Factor r= 25 | ||
|---|---|---|
| Preimage | Multiply by r | Image |
| A(0,5) | ( 2/5(0), 2/5(5)) | A'(0,2) |
| B(- 5,0) | ( 2/5(- 5), 2/5(0)) | B'(- 2,0) |
| C(5,- 5) | ( 2/5(5), 2/5(- 5)) | C'(2, - 2) |
The scale factor is 25. Note that this is the reciprocal of the scale factor our friend came up with. 2/5 ✓ 5/2 * Our friend most likely mistook △ ABC for image and △ A'B'C' for preimage. This would explain why the values in the numerator and denominator of the scale factor were reversed.