Answer :
To determine which statements are true regarding the two triangles after the transformations, let's go through the logical steps one by one.
1. Scaling Transformations:
- We first apply the scaling transformation [tex]\( D_{ O , 2} \)[/tex] (scaling by a factor of 2) to each vertex of the original triangle [tex]\( \triangle LMN \)[/tex].
- Then, we apply the scaling transformation [tex]\( D_{ O , 0.75} \)[/tex] (scaling by a factor of 0.75) to the results obtained from the first scaling.
2. Coordinate Changes:
- For the initial coordinates:
- [tex]\( L = (-3, 1.5) \)[/tex]
- [tex]\( N = (3, -1.5) \)[/tex]
- Assume [tex]\( M = M_{\text{initial}} \)[/tex]. Here we treat [tex]\( M_{\text{initial}} \)[/tex] as having the same coordinates as [tex]\( L \)[/tex] due to placeholder values.
3. Applying [tex]\( D_{ O , 2} \)[/tex]:
- [tex]\( L' = (-3 \times 2, 1.5 \times 2) = (-6, 3) \)[/tex]
- [tex]\( N' = (3 \times 2, -1.5 \times 2) = (6, -3) \)[/tex]
- Since [tex]\( M \)[/tex] is treated initially the same as [tex]\( L \)[/tex]:
- [tex]\( M' = (-6, 3) \)[/tex]
4. Applying [tex]\( D_{ O , 0.75} \)[/tex]:
- [tex]\( L'' = (-6 \times 0.75, 3 \times 0.75) = (-4.5, 2.25) \)[/tex]
- [tex]\( N'' = (6 \times 0.75, -3 \times 0.75) = (4.5, -2.25) \)[/tex]
- [tex]\( M'' = (-4.5, 2.25) \)[/tex] (as [tex]\( M' \)[/tex] was identical to [tex]\( L' \)[/tex], for this exercise)
5. Statements Analysis:
- [tex]\( \angle M \simeq \angle M'' \)[/tex]: True, because scaling transformations preserve angles.
- [tex]\( \triangle LMN \sim \triangle L'' M'' N'' \)[/tex]: True, because similarity is preserved under uniform scaling.
- [tex]\( \triangle LMN \cong \triangle L'' M'' N'' \)[/tex]: False, because congruent triangles must have the same size and shape, and scaling changes the size.
- The coordinates of vertex [tex]\( L'' \)[/tex] are (-3, 1.5): False, the coordinates of [tex]\( L'' \)[/tex] are indeed (-4.5, 2.25).
- The coordinates of vertex [tex]\( N'' \)[/tex] are (3, -1.5): False, the coordinates of [tex]\( N'' \)[/tex] are (4.5, -2.25).
- The coordinates of vertex [tex]\( M'' \)[/tex] are (1.5, -1.5): False, the coordinates of [tex]\( M'' \)[/tex] are (-4.5, 2.25).
Therefore, the statements that are true are:
- [tex]\( \angle M \simeq \angle M'' \)[/tex]
- [tex]\( \triangle LMN \sim \triangle L'' M'' N'' \)[/tex]
1. Scaling Transformations:
- We first apply the scaling transformation [tex]\( D_{ O , 2} \)[/tex] (scaling by a factor of 2) to each vertex of the original triangle [tex]\( \triangle LMN \)[/tex].
- Then, we apply the scaling transformation [tex]\( D_{ O , 0.75} \)[/tex] (scaling by a factor of 0.75) to the results obtained from the first scaling.
2. Coordinate Changes:
- For the initial coordinates:
- [tex]\( L = (-3, 1.5) \)[/tex]
- [tex]\( N = (3, -1.5) \)[/tex]
- Assume [tex]\( M = M_{\text{initial}} \)[/tex]. Here we treat [tex]\( M_{\text{initial}} \)[/tex] as having the same coordinates as [tex]\( L \)[/tex] due to placeholder values.
3. Applying [tex]\( D_{ O , 2} \)[/tex]:
- [tex]\( L' = (-3 \times 2, 1.5 \times 2) = (-6, 3) \)[/tex]
- [tex]\( N' = (3 \times 2, -1.5 \times 2) = (6, -3) \)[/tex]
- Since [tex]\( M \)[/tex] is treated initially the same as [tex]\( L \)[/tex]:
- [tex]\( M' = (-6, 3) \)[/tex]
4. Applying [tex]\( D_{ O , 0.75} \)[/tex]:
- [tex]\( L'' = (-6 \times 0.75, 3 \times 0.75) = (-4.5, 2.25) \)[/tex]
- [tex]\( N'' = (6 \times 0.75, -3 \times 0.75) = (4.5, -2.25) \)[/tex]
- [tex]\( M'' = (-4.5, 2.25) \)[/tex] (as [tex]\( M' \)[/tex] was identical to [tex]\( L' \)[/tex], for this exercise)
5. Statements Analysis:
- [tex]\( \angle M \simeq \angle M'' \)[/tex]: True, because scaling transformations preserve angles.
- [tex]\( \triangle LMN \sim \triangle L'' M'' N'' \)[/tex]: True, because similarity is preserved under uniform scaling.
- [tex]\( \triangle LMN \cong \triangle L'' M'' N'' \)[/tex]: False, because congruent triangles must have the same size and shape, and scaling changes the size.
- The coordinates of vertex [tex]\( L'' \)[/tex] are (-3, 1.5): False, the coordinates of [tex]\( L'' \)[/tex] are indeed (-4.5, 2.25).
- The coordinates of vertex [tex]\( N'' \)[/tex] are (3, -1.5): False, the coordinates of [tex]\( N'' \)[/tex] are (4.5, -2.25).
- The coordinates of vertex [tex]\( M'' \)[/tex] are (1.5, -1.5): False, the coordinates of [tex]\( M'' \)[/tex] are (-4.5, 2.25).
Therefore, the statements that are true are:
- [tex]\( \angle M \simeq \angle M'' \)[/tex]
- [tex]\( \triangle LMN \sim \triangle L'' M'' N'' \)[/tex]
