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.