Move shapes using vector notation. Understand translation as sliding without rotation or changing size.
Click on each step to explore translation on coordinates:
Translation means sliding a shape to a new position without rotating, flipping, or changing its size. Every point moves the same distance in the same direction.
Describe translations using vectors written as (x, y). The x-value shows horizontal movement, y-value shows vertical movement.
Add the vector values to each coordinate of the original shape. If a point is at (a, b) and the vector is (x, y), the new point is at (a+x, b+y).
Connect the translated points in the same order as the original shape. The new shape should be identical to the original but in a different position.
Translation is one of the fundamental transformations in geometry, allowing us to move shapes precisely using coordinate mathematics.
Draw a shape, then use the controls to translate it!
Animators use translation to move characters and objects smoothly across screens in movies, games, and apps.
Navigation systems calculate your movement using coordinate translations to update your position on digital maps.
Robots use translation mathematics to plan precise movements and navigate through physical spaces.
Original triangle: A(1, 2), B(3, 2), C(2, 4)
Translation vector: (3, 1)
Solution:
Result: New triangle A'B'C' is 3 units right and 1 unit up from the original.
Original square: P(2, 3), Q(4, 3), R(4, 5), S(2, 5)
Translation vector: (-2, -1)
Solution:
Result: New square P'Q'R'S' is 2 units left and 1 unit down from the original.
Start: Point M(1, 1)
First translation: (2, 3) → M₁(3, 4)
Second translation: (-1, 2) → M₂(2, 6)
Combined effect:
Key insight: Multiple translations can be combined by adding their vectors together.
Problem: Using (+2, -3) when the movement is actually left 2, up 3.
Solution: Remember: positive x = right, negative x = left, positive y = up, negative y = down.
Problem: Subtracting vector values instead of adding them to coordinates.
Solution: Always ADD the vector components to the original coordinates: (x, y) + (a, b) = (x+a, y+b).
Problem: Expecting the shape to rotate, flip, or change size during translation.
Solution: Translation only moves - the shape stays identical in every way except position.
You have mastered translation on coordinates! Here's what you can now do:
Use (x, y) vectors to describe precise movements on coordinate grids
Move geometric figures to new positions while preserving their properties
Add vector components to original coordinates to find translated positions
Recognize that translation preserves size, shape, angles, and orientation
You've mastered translation on coordinates
You can now move shapes precisely using mathematical vectors and understand how position changes while properties stay the same!
Ready to learn about reflection? In Chapter 8.4, you'll discover how to flip shapes across the x-axis and y-axis, creating mirror images using coordinates!