Question

Answer this question please

Answer 1

Reflection only

The mirror line must pass through one of the sides of the triangle.

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Step-by-step explanation:

The term "invariant" means "does not vary" i.e. "does not change".

An invariant point is fixed in place.

Here are the list of geometric transformations

1. Translation (aka shifting)
2. Rotation
3. Reflection
4. Dilation

Some geometry textbooks will mention "glide reflection", but that's really just a combination of translation and reflection. That means we can ignore it.

Let's go through each item to see if we can get two vertices to stay glued in their spot.

1. With translations, it's impossible to have invariant points. Every point will change location. Therefore, we cannot have two invariant vertices for any translation. We cross translation off the list.
2. Rotations are a little better. We have one point that doesn't move: the center of rotation. Unfortunately every other point that isn't the center will rotate around the center (and hence change location). We cross rotations off the list.
3. Reflections are the transformation that will be able to hold two vertices fixed in place. Any point on the mirror line will not change location. If we draw the mirror line through a side of the triangle, then it will guarantee two vertices will stay in place. This is why reflection is the only answer.
4. Dilations are similar to rotations in that they have one invariant point. The center of dilation stays where it is, but everything else moves closer to the center (if the scale factor is between 0 and 1) or moves away from the center (if the scale factor is larger than 1). We cross dilations off the list.

In short we have eliminated: translations, rotations, and dilations. The only thing that works is reflections