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. c

To translate an entire triangle given the coordinates of its vertices and the coordinates of one vertex of its image under a translation, follow these steps:<br /><br />1. **Identify the Translation Vector:**<br /> - Let the original vertex be \( A(x_1, y_1) \).<br /> - Let the translated vertex be \( A'(x_2, y_2) \).<br /> - The translation vector \( \vec{T} \) can be found by calculating:<br /> \[<br /> \vec{T} = (x_2 - x_1, y_2 - y_1)<br /> \]<br /><br />2. **Apply the Translation Vector to All Vertices:**<br /> - For each vertex \( B(x_3, y_3) \) of the triangle, apply the translation vector \( \vec{T} \) to find the translated vertex \( B' \):<br /> \[<br /> B'(x_3', y_3') = (x_3 + (x_2 - x_1), y_3 + (y_2 - y_1))<br /> \]<br /> - Repeat this for all vertices \( C \) and \( D \) of the triangle.<br /><br />3. **Plot the Translated Vertices:**<br /> - Once you have the translated coordinates for all vertices, plot these new points on the coordinate plane.<br /><br />4. **Connect the Translated Vertices:**<br /> - Draw lines between the translated vertices to form the translated triangle.<br /><br />### Example:<br />Suppose you have a triangle with vertices \( A(1, 2) \), \( B(3, 4) \), and \( C(5, 6) \). You are given that the vertex \( A \) translates to \( A'(4, 5) \).<br /><br />1. **Find the Translation Vector:**<br /> \[<br /> \vec{T} = (4 - 1, 5 - 2) = (3, 3)<br /> \]<br /><br />2. **Translate Each Vertex:**<br /> - For \( B(3, 4) \):<br /> \[<br /> B'(x_3', y_3') = (3 + 3, 4 + 3) = (6, 7)<br /> \]<br /> - For \( C(5, 6) \):<br /> \[<br /> C'(x_4', y_4') = (5 + 3, 6 + 3) = (8, 9)<br /> \]<br /><br />3. **Plot the Translated Vertices:**<br /> - \( A'(4, 5) \)<br /> - \( B'(6, 7) \)<br /> - \( C'(8, 9) \)<br /><br />4. **Connect the Translated Vertices:**<br /> - Draw lines between \( A'(4, 5) \), \( B'(6, 7) \), and \( C'(8, 9) \) to form the translated triangle.<br /><br />By following these steps, you can accurately translate the entire triangle using the given information.