Question

Consider a triangle with vertices at (1, 2), (3, 4), and (2, 6). If it is first dilated with a scale factor of 2, and then translated 3 units to the left and 4 units down, what are the new coordinates of the vertices?

Answer 1

To find the new coordinates of the triangle after dilation and translation, we will follow these steps:

1. **Dilation with a scale factor of 2:**
The formula for dilation is given by \( (x, y) \rightarrow (kx, ky) \) where \( k \) is the scale factor. For a scale factor of 2,
- \( (1, 2) \) dilates to \( (1 \cdot 2, 2 \cdot 2) = (2, 4) \)
- \( (3, 4) \) dilates to \( (3 \cdot 2, 4 \cdot 2) = (6, 8) \)
- \( (2, 6) \) dilates to \( (2 \cdot 2, 6 \cdot 2) = (4, 12) \)

After dilation, the vertices of the triangle are:
- \( (2, 4) \)
- \( (6, 8) \)
- \( (4, 12) \)

2. **Translation (3 units left and 4 units down):**
The translation rule for translating a point \( (x, y) \) to the left by 3 units and down by 4 units is given by \( (x, y) \rightarrow (x - 3, y - 4) \).
Apply this translation to the new coordinates:
- For \( (2, 4) \):
\[
(2 - 3, 4 - 4) = (-1, 0)
\]
- For \( (6, 8) \):
\[
(6 - 3, 8 - 4) = (3, 4)
\]
- For \( (4, 12) \):
\[
(4 - 3, 12 - 4) = (1, 8)
\]

The final coordinates of the vertices after dilation and translation are:
- Vertex 1: \( (-1, 0) \)
- Vertex 2: \( (3, 4) \)
- Vertex 3: \( (1, 8) \)

Thus, the new coordinates of the vertices of the triangle are:
\[
\boxed{(-1, 0), (3, 4), (1, 8)}
\]