Certainly! Here's a clear explanation of how to translate the entire triangle when you have:
- The coordinates of the original triangle's three vertices, and
- The coordinates of only one vertex of the translated image.
How to Translate the Entire Triangle
Step 1: Identify the given points
- Let the original triangle have vertices A(x1,y1)A(x_1,y_1)A(x1,y1), B(x2,y2)B(x_2,y_2)B(x2,y2), and C(x3,y3)C(x_3,y_3)C(x3,y3).
- You are given the image of one vertex after translation, say A′(x1′,y1′)A'(x_1',y_1')A′(x1′,y1′).
Step 2: Find the translation vector
- A translation moves every point by the same amount.
- The translation vector v⃗\vec{v}v is the difference between the image vertex and the original vertex:
v⃗=(x1′−x1,y1′−y1)\vec{v}=(x_1'-x_1,\quad y_1'-y_1)v=(x1′−x1,y1′−y1)
- This vector tells you how far and in what direction the triangle has moved.
Step 3: Apply the translation vector to the other vertices
- To find the image of vertex BBB, add the translation vector to BBB:
B′=(x2+(x1′−x1),y2+(y1′−y1))B'=(x_2+(x_1'-x_1),\quad y_2+(y_1'-y_1))B′=(x2+(x1′−x1),y2+(y1′−y1))
- Similarly, for vertex CCC:
C′=(x3+(x1′−x1),y3+(y1′−y1))C'=(x_3+(x_1'-x_1),\quad y_3+(y_1'-y_1))C′=(x3+(x1′−x1),y3+(y1′−y1))
Step 4: Write the coordinates of the translated triangle
- The translated triangle has vertices A′(x1′,y1′)A'(x_1',y_1')A′(x1′,y1′), B′B'B′, and C′C'C′.
Summary
- Find the translation vector by subtracting the original vertex coordinates from the translated vertex coordinates.
- Add this vector to the other vertices of the original triangle.
- The resulting points give you the coordinates of the entire translated triangle.
If you'd like, I can also provide a numerical example or help with a specific set of points!
