Do the ratios have to have the same scale factor to be a dilation? My friend says yes, or else the scale factor would be useless. I on the other hand say no because it's still a dilation as it's an reduction.

Here's the instructions: "For Exercises 5-8, tell whether one figure is a dilation of the other or not. If one figure is a dilation of the other, tell whether it is an enlargement or a reduction. Explain your reasoning."

And here's the problem I answered to be a reduction: "Triangle R'S'T' has sides of 3 cm, 4 cm, and 5 cm. Triangle RST has sides of 12 cm, 6 cm, and 25 cm."

Also does my explanation sound plausible? I answered reduction and my reasoning was "because the scale factor of each coordinate[/ratio] is less than 1 and more than 0 (decimal)".

your friend is right, you have failed to see that a dilation with a scale factor less than one is a reduction.

I guess you didn't see my second post before answering. But then, I did say in my first post, " I on the other hand say no because it's still a dilation as it's an reduction." Sorry for late reply because I had to catch the bus (I was using the library's WIFI to post this question & I don't have data to use a network).

I forgot to include the scale factors I got for each ratio: image/preimage. So for 3/12 it is 0.25, 4/16 (typo in first post) is 0.25, and 5/25 is 0.20(!). The last ratio is what's bothering me. Because one is different from the others, does it still make it a dilation? That's what I was asking.

Where are my manners? I forgot to say thank you for answering my question.

I guess I'll just get answers from an answer key.

To determine whether two figures are dilations of each other, we need to compare the corresponding side lengths. A dilation occurs when one figure is an exact scaled copy of the other.

In your case, you correctly identified that Triangle R'S'T' is a reduction (a type of dilation) of Triangle RST. This is because all corresponding side lengths in Triangle R'S'T' are smaller than the corresponding side lengths in Triangle RST. The ratios of the corresponding side lengths are not required to have the same scale factor.

For example, let's compare the side lengths of Triangle R'S'T' and Triangle RST:
- The ratio of the corresponding first side lengths is 3 cm / 12 cm = 1/4.
- The ratio of the corresponding second side lengths is 4 cm / 6 cm = 2/3.
- The ratio of the corresponding third side lengths is 5 cm / 25 cm = 1/5.

As you can see, the scale factors for the side lengths are different, yet Triangle R'S'T' is still a dilation of Triangle RST because it is an exact scaled-down copy. The scale factor for each side length doesn't have to be the same; it just needs to be proportional.

Therefore, your explanation that Triangle R'S'T' is a reduction is correct.