Answer :
Let's solve this step-by-step to determine the transformation type and the coordinates of the vertices after the reflections.
### Step 1: Identify the type of transformation
Since the problem appears to be describing a sequence of transformations leading from [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A''B''C''\)[/tex], and it specifically mentions a reflection, we need to infer the initial transformation (the one mapping [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A'B'C'\)[/tex]).
### Step 2: Reflection across the line [tex]\(x = -2\)[/tex]
To better understand which vertex [tex]\(B''\)[/tex] will have the same coordinates as [tex]\(B'\)[/tex], reflect the vertices of [tex]\(\triangle A'B'C'\)[/tex] across the line [tex]\(x = -2\)[/tex].
### Process of reflection:
- When you reflect a point [tex]\( (x, y) \)[/tex] over the vertical line [tex]\(x = -2\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate of the reflected point becomes [tex]\( -2 - (x - (-2)) \)[/tex] which simplifies to [tex]\(-4 - x\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains unchanged.
The vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex] of [tex]\(\triangle A'B'C'\)[/tex] when reflected over the line [tex]\(x = -2\)[/tex]:
- Vertex A': [tex]\((x_1, y_1) \rightarrow (-4 - x_1, y_1)\)[/tex]
- Vertex B': [tex]\((x_2, y_2) \rightarrow (-4 - x_2, y_2)\)[/tex]
- Vertex C': [tex]\((x_3, y_3) \rightarrow (-4 - x_3, y_3)\)[/tex]
### Step 3: Analyze the coordinates after reflection
We need to find which vertex of [tex]\(\triangle A'' B'' C''\)[/tex] will match the coordinates of [tex]\(B'\)[/tex]:
- If we assume that a particular vertex has its reflection producing coordinates that match another point, we should consider the exact match situations.
If vertex [tex]\(B'\)[/tex] gets mapped to itself (has the same coordinates after reflection), it generally happens when the reflecting line [tex]\( x = -2 \)[/tex] and the reflected vertex have the same distance from this line.
### Step 4: Determine the repeated coordinates:
To ensure we match the process correctly, follow the problem statement regarding which vertex [tex]\(B''\)[/tex] will coincide with [tex]\(B'\)[/tex]:
- Given condition is perfect in scenarios of symmetry or specific points.
### Summary conclusion:
Without changing specifics, if any vertex [tex]\(B''\)[/tex] is identical to [tex]\(B'\)[/tex], originally this type of matching happens only due to mapping processes and should be considered vertex-wise identical after reflections.
## The Answers:
- The type of transformation that maps [tex]\(\triangle ABC\)[/tex] onto [tex]\(\triangle A'B'C'\)[/tex] is a Reflection\ (or a more specific function of matching reflection like ref).
- When [tex]\(\triangle A'B'C'\)[/tex] is reflected across the line [tex]\(x = -2\)[/tex] to form [tex]\(\triangle A''B''C''\)[/tex], vertex [tex]\(A' = (x_2, y_2)\)[/tex] of [tex]\(\triangle A'' B'' C''\)[/tex] will have the same coordinates as [tex]\(B'\)[/tex].
Note: Always interpreting correct steps ensuring no mismatch.
### Step 1: Identify the type of transformation
Since the problem appears to be describing a sequence of transformations leading from [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A''B''C''\)[/tex], and it specifically mentions a reflection, we need to infer the initial transformation (the one mapping [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A'B'C'\)[/tex]).
### Step 2: Reflection across the line [tex]\(x = -2\)[/tex]
To better understand which vertex [tex]\(B''\)[/tex] will have the same coordinates as [tex]\(B'\)[/tex], reflect the vertices of [tex]\(\triangle A'B'C'\)[/tex] across the line [tex]\(x = -2\)[/tex].
### Process of reflection:
- When you reflect a point [tex]\( (x, y) \)[/tex] over the vertical line [tex]\(x = -2\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate of the reflected point becomes [tex]\( -2 - (x - (-2)) \)[/tex] which simplifies to [tex]\(-4 - x\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains unchanged.
The vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex] of [tex]\(\triangle A'B'C'\)[/tex] when reflected over the line [tex]\(x = -2\)[/tex]:
- Vertex A': [tex]\((x_1, y_1) \rightarrow (-4 - x_1, y_1)\)[/tex]
- Vertex B': [tex]\((x_2, y_2) \rightarrow (-4 - x_2, y_2)\)[/tex]
- Vertex C': [tex]\((x_3, y_3) \rightarrow (-4 - x_3, y_3)\)[/tex]
### Step 3: Analyze the coordinates after reflection
We need to find which vertex of [tex]\(\triangle A'' B'' C''\)[/tex] will match the coordinates of [tex]\(B'\)[/tex]:
- If we assume that a particular vertex has its reflection producing coordinates that match another point, we should consider the exact match situations.
If vertex [tex]\(B'\)[/tex] gets mapped to itself (has the same coordinates after reflection), it generally happens when the reflecting line [tex]\( x = -2 \)[/tex] and the reflected vertex have the same distance from this line.
### Step 4: Determine the repeated coordinates:
To ensure we match the process correctly, follow the problem statement regarding which vertex [tex]\(B''\)[/tex] will coincide with [tex]\(B'\)[/tex]:
- Given condition is perfect in scenarios of symmetry or specific points.
### Summary conclusion:
Without changing specifics, if any vertex [tex]\(B''\)[/tex] is identical to [tex]\(B'\)[/tex], originally this type of matching happens only due to mapping processes and should be considered vertex-wise identical after reflections.
## The Answers:
- The type of transformation that maps [tex]\(\triangle ABC\)[/tex] onto [tex]\(\triangle A'B'C'\)[/tex] is a Reflection\ (or a more specific function of matching reflection like ref).
- When [tex]\(\triangle A'B'C'\)[/tex] is reflected across the line [tex]\(x = -2\)[/tex] to form [tex]\(\triangle A''B''C''\)[/tex], vertex [tex]\(A' = (x_2, y_2)\)[/tex] of [tex]\(\triangle A'' B'' C''\)[/tex] will have the same coordinates as [tex]\(B'\)[/tex].
Note: Always interpreting correct steps ensuring no mismatch.
