Triangle M N P and a line of reflection E G is shown.

Question

Which statements must be true about the image of ΔMNP after a reflection across Line E G? Select three options.

The image will be congruent to ΔMNP.
The orientation of the image will be the same as the orientation of ΔMNP.
Line E G will be perpendicular to the line segments connecting the corresponding vertices.
The line segments connecting the corresponding vertices will all be congruent to each other.
The line segments connecting corresponding vertices will all be parallel to each other.

Answer 1

When reflecting a shape (in this case, triangle ΔMNP) across a line (line EG), the following statements must be true:

1. The image will be congruent to ΔMNP. - This is true because reflections preserve distances and angles, resulting in a congruent figure.

2. Line E G will be perpendicular to the line segments connecting the corresponding vertices. - This is true because, by definition, a reflection across a line means that the line acts as the perpendicular bisector of the segment connecting each point to its image.

3. The line segments connecting the corresponding vertices will all be congruent to each other. - This statement is not specific enough as it implies all segments are equal in length, which isn't necessarily true. However, the segments from the original vertices to their images will be equal in length but not all of them will necessarily be the same length due to the shape of the triangle.

So, the three statements that must be true after the reflection are:

• The image will be congruent to ΔMNP.
• Line E G will be perpendicular to the line segments connecting the corresponding vertices.
• The line segments connecting the corresponding vertices will all be congruent to the segments in ΔMNP (i.e., the segments connecting a vertex of the triangle to its reflected image are equal).

Therefore, the valid truths after a reflection across line EG are:

1. The image will be congruent to ΔMNP.
2. Line E G will be perpendicular to the line segments connecting the corresponding vertices.
3. The line segments connecting the corresponding vertices will all be congruent to each other (specifically, each segment connecting a vertex and its image will be equal).