Connect with knowledgeable experts and enthusiasts on IDNLearn.com. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
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.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.
