Rules for Translations on a Coordinate Plane

📚 Understanding Translations on a Coordinate Plane

A translation is a type of transformation that slides a figure from one location to another on the coordinate plane without changing its size, shape, or orientation. Think of it like picking up a puzzle piece and placing it somewhere else on the board without rotating it.

📝 Key Rules for Translations

• 📏 Vectors Define Movement: Translations are defined by a translation vector, often written as $ \langle a, b \rangle $. The 'a' represents the horizontal shift (positive for right, negative for left), and the 'b' represents the vertical shift (positive for up, negative for down).
• ➕ Adding to Coordinates: To translate a point $ (x, y) $ by the vector $ \langle a, b \rangle $, you add 'a' to the x-coordinate and 'b' to the y-coordinate. The translated point becomes $ (x + a, y + b) $.
• 📍 Applying to All Points: If you're translating a shape, apply the translation vector to every vertex (corner point) of the shape. Then, connect the translated vertices to create the translated shape.
• 🔄 Preservation of Shape: Translations are rigid transformations (also called isometries). This means the size and shape of the figure remain the same. Only the position changes.
• 📈 Parallel Lines: Translations preserve parallelism. If two lines in the original figure are parallel, their corresponding lines in the translated figure will also be parallel.
• 🧭 Orientation Remains: The orientation (clockwise or counterclockwise order of vertices) of the figure is preserved during a translation.
• 💡 Combining Translations: You can perform multiple translations in a row. The overall translation vector is simply the sum of the individual translation vectors. For example, translating by $ \langle 1, 2 \rangle $ and then by $ \langle -3, 1 \rangle $ is the same as translating by $ \langle -2, 3 \rangle $.

✏️ Example: Translating a Triangle

Let's translate triangle ABC with vertices A(1, 1), B(3, 1), and C(1, 4) by the vector $ \langle 2, -3 \rangle $.

• 📍 A'(1+2, 1-3) = A'(3, -2)
• 📍 B'(3+2, 1-3) = B'(5, -2)
• 📍 C'(1+2, 4-3) = C'(3, 1)

So, the translated triangle A'B'C' has vertices A'(3, -2), B'(5, -2), and C'(3, 1).

✍️ Practice Quiz

Translate each point or shape using the given translation vector.

1. 🔢 Point P(2, 3) by $ \langle -1, 4 \rangle $
2. 📐 Point Q(-1, -2) by $ \langle 3, -2 \rangle $
3. 🟩 Square ABCD with A(0, 0), B(2, 0), C(2, 2), D(0, 2) by $ \langle 1, 1 \rangle $
4. ⭐ Triangle EFG with E(-2, 1), F(0, 1), G(-1, 3) by $ \langle 2, -1 \rangle $
5. ♦️ Rectangle HIJK with H(1, -1), I(4, -1), J(4, -3), K(1, -3) by $ \langle -2, 3 \rangle $
6. 🛑 Line segment LM with L(5, 0), M(5, 5) by $ \langle -5, -2 \rangle $
7. 🌀 Circle with center (0,0) and radius 3 by $ \langle 4,0 \rangle $