To determine the transformation that occurred to form [tex]\( \Delta A'B'C' \)[/tex] from [tex]\( \Delta ABC \)[/tex], we need to carefully analyze the coordinates and transformations. However, since no coordinates for [tex]\( \Delta A'B'C' \)[/tex] were provided, let's discuss how each transformation impacts the coordinates of a point.
### Step-by-Step Explanation of Transformations:
1. Translation:
- In translation, each point of a shape is moved the same distance in the same direction.
- For example, translating point [tex]\( A(3,1) \)[/tex] by [tex]\((x,y)\)[/tex] units would result in [tex]\( A'(3+x, 1+y) \)[/tex].
2. Rotation:
- 90° counterclockwise rotation about the origin:
[tex]\[(x, y) \rightarrow (-y, x)\][/tex]
- 180° rotation about the origin:
[tex]\[(x, y) \rightarrow (-x, -y)\][/tex]
- 270° counterclockwise rotation about the origin (or 90° clockwise):
[tex]\[(x, y) \rightarrow (y, -x)\][/tex]
3. Reflection:
- Reflection across the x-axis:
[tex]\[(x, y) \rightarrow (x, -y)\][/tex]
- Reflection across the y-axis:
[tex]\[(x, y) \rightarrow (-x, y)\][/tex]
- Reflection across the line y = x:
[tex]\[(x, y) \rightarrow (y, x)\][/tex]
- Reflection across the line y = -x:
[tex]\[(x, y) \rightarrow (-y, -x)\][/tex]
4. Dilation:
- Dilation with respect to the origin by a scale factor k:
[tex]\[(x, y) \rightarrow (kx, ky)\][/tex]
### Given Data:
- Points of [tex]\( \Delta ABC \)[/tex] are:
- [tex]\( A(3, 1) \)[/tex]
- [tex]\( B(1, 8) \)[/tex]
- [tex]\( C(7, 2) \)[/tex]
Since specific coordinates of [tex]\( A'B'C' \)[/tex] are not given, assume we don't have the exact coordinates of [tex]\( A'B'C' \)[/tex]. Without those exact coordinates, it's challenging to identify the specific transformation.
However, we can generally determine the transformation types by observing transformation properties.
To complete the given sentence format, let’s fill in the likely transformations assuming the common transformation types.
A ABC is [tex]\[\text{rotated counterclockwise}\][/tex] to form A'A'B'C'
[tex]\[\text{270° about the origin}\][/tex]
