Question

A triangle is reflected over the y-axis and then translated 3 units up. If the original coordinates of the triangle's vertices are (2, 1), (4, 3), and (3, 5), what are the coordinates of the vertices after the transformations?
A
A dilation with a scale factor of 0.5.
B
A translation 6 units up.
C
A rotation of 45 degrees.
D
A reflection over the y-axis.

Answer 1

To determine the coordinates of the triangle's vertices after it is reflected over the y-axis and translated 3 units up, we can follow these steps:

1. **Reflection over the y-axis**: When a point \((x, y)\) is reflected over the y-axis, its coordinates become \((-x, y)\). We apply this transformation to each vertex of the triangle.

- For vertex \((2, 1)\):
\[
(2, 1) \rightarrow (-2, 1)
\]
- For vertex \((4, 3)\):
\[
(4, 3) \rightarrow (-4, 3)
\]
- For vertex \((3, 5)\):
\[
(3, 5) \rightarrow (-3, 5)
\]

After reflection, the new coordinates of the vertices are:
- Vertex 1: \((-2, 1)\)
- Vertex 2: \((-4, 3)\)
- Vertex 3: \((-3, 5)\)

2. **Translation 3 units up**: To translate a point \(3\) units up, we add \(3\) to the y-coordinate. We apply this transformation to each of the reflected vertices.

- For vertex \((-2, 1)\):
\[
(-2, 1) \rightarrow (-2, 1 + 3) = (-2, 4)
\]
- For vertex \((-4, 3)\):
\[
(-4, 3) \rightarrow (-4, 3 + 3) = (-4, 6)
\]
- For vertex \((-3, 5)\):
\[
(-3, 5) \rightarrow (-3, 5 + 3) = (-3, 8)
\]

After translation, the final coordinates of the vertices are:
- Vertex 1: \((-2, 4)\)
- Vertex 2: \((-4, 6)\)
- Vertex 3: \((-3, 8)\)

So, the coordinates of the triangle's vertices after the transformations are:
- \((-2, 4)\)
- \((-4, 6)\)
- \((-3, 8)\)