Question

You are given the coordinates of a triangle and coordinates for only one of the vertices of its image under a translation. Explain how to translate the entire triangle.

Answer 1

To translate an entire triangle based on the provided vertex coordinates of the triangle and the coordinates of its translated image vertex, you can follow these steps:

1. **Identify the given information**:
- Let the vertices of the triangle be \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).
- Let the coordinates of the vertex that you have of the image triangle be \( A'(x'_1, y'_1) \).

2. **Find the translation vector**:
- The translation vector can be computed by finding the difference between the image vertex and the original vertex.
- For vertex \( A \), the translation vector \( \vec{T} \) is given by:
\[
\vec{T} = (x'_1 - x_1, y'_1 - y_1)
\]

3. **Apply the translation to the entire triangle**:
- To translate the entire triangle, add the translation vector \( \vec{T} \) to each vertex of the original triangle:
- For vertex \( A \):
\[
A' = A + \vec{T} = (x_1 + (x'_1 - x_1), y_1 + (y'_1 - y_1)) = (x'_1, y'_1)
\]
- For vertex \( B \):
\[
B' = B + \vec{T} = (x_2 + (x'_1 - x_1), y_2 + (y'_1 - y_1))
\]
- For vertex \( C \):
\[
C' = C + \vec{T} = (x_3 + (x'_1 - x_1), y_3 + (y'_1 - y_1))
\]

4. **Final translated vertices**:
- The new vertices of the translated triangle will be:
- \( A' = (x'_1, y'_1) \)
- \( B' = (x_2 + (x'_1 - x_1), y_2 + (y'_1 - y_1)) \)
- \( C' = (x_3 + (x'_1 - x_1), y_3 + (y'_1 - y_1)) \)

By following these steps, you can successfully translate the entire triangle according to the given vertex coordinates and the corresponding image vertex after the translation.