When analyzing the transformation and relationships between the triangles [tex]$\triangle LMN$[/tex] and [tex]$\triangle L''M''N''$[/tex], we reach the following conclusions:
1. Angles remain the same after transformations (Angle preservation under similarity transformations):
[tex]\[
\angle M = \angle M''
\][/tex]
This statement is true because the angles of the triangles remain the same when a transformation involving scaling is applied.
2. Triangles are similar:
[tex]\[
\triangle LMN \sim \triangle L''M''N''
\][/tex]
This statement is true. Similar transformations, such as scaling, do not alter the shape of the triangles but only their size. Thus, the triangles are similar.
3. Triangles are congruent:
[tex]\[
\triangle LMN = \triangle L''M''N''
\][/tex]
This statement is false. Although the triangles remain similar, they differ in size due to the scaling factors applied. Therefore, they are not congruent since congruence requires identical size and shape.
4. Coordinates of vertex [tex]$L''$[/tex]:
[tex]\[
L'' = (-3, 1.5)
\][/tex]
This statement is true. According to the transformations, the coordinates of the vertex [tex]$L''$[/tex] after the first and second scaling transformations are indeed [tex]$(-3, 1.5)$[/tex].
5. Coordinates of vertex [tex]$N''$[/tex]:
[tex]\[
N'' = (3, -1.5)
\][/tex]
This statement is true. Based on the transformations, the coordinates of vertex [tex]$N''$[/tex] after scaling transformations are [tex]$(3, -1.5)$[/tex].
6. Coordinates of vertex [tex]$M''$[/tex]:
[tex]\[
M'' = (1.5, -1.5)
\][/tex]
This statement is true. After the scaling transformations, the coordinates of vertex [tex]$M''$[/tex] are [tex]$(1.5, -1.5)$[/tex].
So, the correct statements regarding the triangles [tex]$\triangle LMN$[/tex] and [tex]$\triangle L''M''N''$[/tex] are:
- [tex]\(\angle M = \angle M''\)[/tex]
- [tex]\(\triangle LMN \sim \triangle L''M''N''\)[/tex]
- The coordinates of vertex [tex]$L''$[/tex] are [tex]$(-3, 1.5)$[/tex].
- The coordinates of vertex [tex]$N''$[/tex] are [tex]$(3, -1.5)$[/tex].
- The coordinates of vertex [tex]$M''$[/tex] are [tex]$(1.5, -1.5)$[/tex].
