Translations
What is a Translation?
A translation is a way of moving a shape from one position to another without rotating it or changing its size. Every point on the shape moves by the same distance in the same direction. Think of it like sliding the shape across the grid—it stays the same size and orientation; it just ends up in a new location.
The key thing to remember is that the object and image are congruent (exactly the same size and shape), and any point on the mirror line doesn't move at all—these are called invariant points.
Translation Vectors
We describe a translation using a column vector, which tells us exactly how far and in which direction to move the shape.
Vector format: (x, y)
• x = horizontal distance (negative = left, positive = right)
• y = vertical distance (negative = down, positive = up)
💡 Tip: Remember that the vector tells you how each point moves. A vector of (3, -1) means "go 3 right and 1 down"—this applies to every single point on the shape.
How to Translate a Shape
Step 1: Interpret the translation vector. For example, (−4, 5) means 4 left and 5 up.
Step 2: Move each vertex on the original shape by counting the grid squares. Start with one vertex, then move to the next.
Step 3: Connect the new vertices together to draw the translated image.
Step 4: Label the new shape (usually with a prime, like A').
⚠️ Common Mistake: Students sometimes confuse the vector with "the gap between the shapes." The vector is how the shape moves, not the distance between them!
How to Describe a Translation
If you're given two shapes and asked to describe the transformation:
Step 1: Pick a point on the original shape and find the corresponding point on the image.
Step 2: Count how many squares right or left you need to go (this is your x value).
Step 3: Count how many squares up or down you need to go (this is your y value).
Step 4: Write your answer as: "A translation by vector (x, y)"
Practice Question 1: Translate triangle ABC with vertices A(1, 2), B(3, 2), and C(2, 4) using the vector (2, -3). Write down the coordinates of the image.
Reversing a Translation
If a shape has been translated by vector (a, b), you can move it back to its original position using the vector (−a, −b). You simply reverse the signs!
Reverse Vector Rule: If the original vector is (x, y), the reverse is (−x, −y)
Example: If translated by (5, −2), reverse it with (−5, 2)