Translations
Read and write column vectors, translate a shape on the coordinate plane, and describe a translation you are shown. Learn which properties survive a slide and which do not.
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Think First
If you slide a shape 3 units right and 2 units up on a grid, what changes and what stays the same? Think about the shape's size, angles, and position.
Translations
A translation slides every point of a shape the same distance in the same direction. Nothing stretches, flips, or turns, it is a pure slide. The shape and its image are congruent, and every side and angle is preserved. We describe translations using a column vector $\binom{a}{b}$, where $a$ is the horizontal shift and $b$ is the vertical shift.
What You'll Master
- Interpret and apply a column vector $\binom{a}{b}$ to translate points and shapes
- Find the image of a shape after a translation and list the image vertices
- Describe a translation given the object and its image (find the vector)
- Understand that translation preserves size, shape, angles, and orientation
Words You Need
⚠ Spot the Trap
In $\binom{a}{b}$, the top number is the horizontal shift and the bottom is the vertical shift. A negative top means move left; a negative bottom means move down. Students often get these backwards or confuse the direction of the sign.
Reading a Column Vector
The column vector $\binom{a}{b}$ means: move $a$ units horizontally and $b$ units vertically.
- $a > 0$: move right; $\quad a < 0$: move left
- $b > 0$: move up; $\quad b < 0$: move down
The rule for translating any point $(x, y)$ under vector $\binom{a}{b}$ is:
$$(x,\; y) \longrightarrow (x + a,\; y + b)$$Example: Translate $(3, 1)$ under $\binom{-2}{4}$:
$$(3, 1) \to (3 + (-2),\; 1 + 4) = (1,\; 5)$$Worked Example, Translating a Triangle
Triangle $ABC$ has vertices $A(1, 0)$, $B(3, 0)$, $C(2, 3)$. Translate it by $\binom{2}{-1}$.
Apply $(x,y) \to (x+2,\; y-1)$ to each vertex:
- $A(1,0) \;\to\; A'(3,\; -1)$
- $B(3,0) \;\to\; B'(5,\; -1)$
- $C(2,3) \;\to\; C'(4,\; 2)$
Plot $A'$, $B'$, $C'$ and connect them. The image is congruent to the original, same size and shape, just in a new position.
Worked Example, Describing a Translation
Point $P(2, 5)$ maps to image $P'(7, 3)$. Find the translation vector.
Method: image coordinates minus object coordinates:
$$\text{Vector} = \binom{x' - x}{y' - y} = \binom{7 - 2}{3 - 5} = \binom{5}{-2}$$Interpretation: 5 right and 2 down. Verify: $(2, 5) \to (2+5,\; 5-2) = (7, 3)$ ✓
Always do image minus object not the other way around.
Common Pitfalls
- Swapping the top and bottom of the vector (horizontal vs vertical)
- Forgetting the sign: $\binom{-3}{2}$ means LEFT 3, UP 2
- Only translating some vertices, not all of them
- When finding a vector, doing object minus image instead of image minus object
Copy This Into Your Book
Translation rule: $(x,\; y) \xrightarrow{\binom{a}{b}} (x+a,\; y+b)$
To find the vector: $\binom{x'-x}{y'-y}$ (image minus object)
Preserved under translation: size, shape, angles, side lengths, orientation. Position changes.
Point $A(2, 4)$ is translated to $A'(5, 1)$. What is the translation vector?
The point $(-1, 3)$ is translated by $\binom{4}{-2}$. What is the image?
Which property of a shape is NOT preserved under a translation?
Vertex $C(4, 5)$ is translated by $\binom{-3}{2}$. Where is $C'$?
$A(1, 2)$ maps to $A'(4, -1)$. Which vector describes this translation?
Q6. Triangle $KLM$ has vertices $K(0, 1)$, $L(4, 1)$, $M(2, 5)$. Translate the triangle by $\binom{-3}{-4}$. List the coordinates of $K'$, $L'$, and $M'$.
Q7. Vertex $P(5, -2)$ maps to $P'(-1, 4)$. Find the translation vector, then find the image of $Q(3, 0)$ under the same translation.
Q8. A translation maps $(2, 5)$ to $(6, 1)$. Find the translation vector, then find the image of $(-1, 3)$ under the same translation.
Show Answers
Q6
Apply $(x-3,\; y-4)$ to each vertex:
$K(0,1) \to K'(-3,-3)$
$L(4,1) \to L'(1,-3)$
$M(2,5) \to M'(-1,1)$
Q7
Vector $= \binom{-1-5}{4-(-2)} = \binom{-6}{6}$
$Q(3,0) \to Q'(3-6,\; 0+6) = Q'(-3,\; 6)$
Q8
Vector $= \binom{6-2}{1-5} = \binom{4}{-4}$
$(-1,3) \to (-1+4,\; 3-4) = (3,\; -1)$
Stretch Challenge
Two translations are applied in sequence: first $\binom{3}{-2}$, then $\binom{-1}{5}$. What single translation vector is equivalent to these two combined? Does the order of applying the translations matter? Justify your answer.