To solve the problem of finding the vertices of [tex]\(\triangle A'B'C'\)[/tex] after [tex]\(\triangle ABC\)[/tex] is reflected over the x-axis and then dilated by a scale factor of 3 about the origin, follow these steps:
### Step 1: Reflect over the x-axis
Reflecting a point [tex]\((x, y)\)[/tex] over the x-axis changes its y-coordinate to [tex]\(-y\)[/tex]. Let's find the reflected points:
1. Point [tex]\(A = (6, 6)\)[/tex]:
- Reflecting over the x-axis: [tex]\(A_{\text{reflected}} = (6, -6)\)[/tex]
2. Point [tex]\(B = (2, -4)\)[/tex]:
- Reflecting over the x-axis: [tex]\(B_{\text{reflected}} = (2, 4)\)[/tex]
3. Point [tex]\(C = (0, 8)\)[/tex]:
- Reflecting over the x-axis: [tex]\(C_{\text{reflected}} = (0, -8)\)[/tex]
So, the vertices after reflection are:
- [tex]\(A_{\text{reflected}} = (6, -6)\)[/tex]
- [tex]\(B_{\text{reflected}} = (2, 4)\)[/tex]
- [tex]\(C_{\text{reflected}} = (0, -8)\)[/tex]
### Step 2: Dilate by a scale factor of 3
Dilating a point [tex]\((x, y)\)[/tex] by a scale factor [tex]\(k\)[/tex] changes it to [tex]\((kx, ky)\)[/tex]. Let's find the dilated points with a scale factor of 3:
1. Point [tex]\(A_{\text{reflected}} = (6, -6)\)[/tex]:
- Dilating by 3: [tex]\(A' = (6 \times 3, -6 \times 3) = (18, -18)\)[/tex]
2. Point [tex]\(B_{\text{reflected}} = (2, 4)\)[/tex]:
- Dilating by 3: [tex]\(B' = (2 \times 3, 4 \times 3) = (6, 12)\)[/tex]
3. Point [tex]\(C_{\text{reflected}} = (0, -8)\)[/tex]:
- Dilating by 3: [tex]\(C' = (0 \times 3, -8 \times 3) = (0, -24)\)[/tex]
So, the vertices after dilation are:
- [tex]\(A' = (18, -18)\)[/tex]
- [tex]\(B' = (6, 12)\)[/tex]
- [tex]\(C' = (0, -24)\)[/tex]
### Conclusion
Thus, the vertices of [tex]\(\triangle A'B'C'\)[/tex] after reflecting [tex]\(\triangle ABC\)[/tex] over the x-axis and then dilating by a scale factor of 3 are:
- [tex]\(A' = (18, -18)\)[/tex]
- [tex]\(B' = (6, 12)\)[/tex]
- [tex]\(C' = (0, -24)\)[/tex]
These are the final coordinates of [tex]\(\triangle A'B'C'\)[/tex].
