Hello there! 👋 It's totally normal to need a bit more clarity on transformations like translations. They're fundamental in geometry, and once you grasp the rules, you'll find them super straightforward! Let's break down how translations work on a coordinate plane.
What is a Translation?
Think of a translation as simply "sliding" a shape or point from one position to another without rotating it, flipping it, or changing its size. Every point of the shape moves the exact same distance in the exact same direction. It's like moving a piece on a chessboard! ♟️
The Golden Rule for Translating a Point
The core of translation lies in how a single point $(x, y)$ moves. When you translate a point, you're essentially adding a certain value to its x-coordinate (for horizontal movement) and a certain value to its y-coordinate (for vertical movement).
Let's say you want to translate a point by $a$ units horizontally and $b$ units vertically.
- If $a$ is positive, the point moves to the right.
- If $a$ is negative, the point moves to the left.
- If $b$ is positive, the point moves up.
- If $b$ is negative, the point moves down.
The rule can be written as:
Given an original point $(x,y)$, its translated image $(x',y')$ will be:
$(x,y) \rightarrow (x+a, y+b)$
Where $(x+a)$ is the new x-coordinate and $(y+b)$ is the new y-coordinate.
Let's Look at Some Examples!
- Translating Right and Up: If you have point $(2,3)$ and you want to translate it 4 units right and 1 unit up, then $a=4$ and $b=1$.
The new point is $(2+4, 3+1) = (6,4)$.
- Translating Left and Down: If you have point $(-1,5)$ and you want to translate it 3 units left and 2 units down, then $a=-3$ and $b=-2$.
The new point is $(-1+(-3), 5+(-2)) = (-4,3)$.
- Translating Only Horizontally: If point $(0,-7)$ is translated 5 units right, then $a=5$ and $b=0$.
The new point is $(0+5, -7+0) = (5,-7)$.
Translating Entire Shapes
When you need to translate an entire shape (like a triangle or a square), you simply apply the exact same translation rule to each of its vertices. Once you've found the new coordinates for all the vertices, you connect them to draw your translated shape! ✨
Key Things to Remember
- Translations are a type of isometry, meaning they preserve the shape's size and angles. Only its position changes.
- The original figure and its translated image are congruent.
- The orientation of the shape does not change. It doesn't get flipped or rotated.
I hope this helps clear things up for your homework and quiz! Practice makes perfect, so try a few more examples. You've got this! 👍