Translations: Step-by-Step Guide
A translation slides a shape without turning it, flipping it, or changing its size. Every point moves the same distance in the same direction. If you translate by (a, b), every point (x, y) moves to (x + a, y + b). The first number tells you how far to move left or right, and the second tells you how far to move up or down.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Find the translation amounts. Find a (the shift left or right) and b (the shift up or down). A problem might say 'translate by (a, b)' or 'move right a units and up b units'.\[\vec{v} = \langle a, b \rangle\]
- Use the rule. Add a to every x-coordinate and add b to every y-coordinate.\[(x, y) \mapsto (x + a,\; y + b)\]
- Find each new point. Use the rule on every corner of the shape and label the new points with primes.\[A(x_A, y_A) \mapsto A'(x_A + a,\; y_A + b)\]
- Check the lengths. The sides and angles of the shape should not change.\[|AA'| = \sqrt{a^2 + b^2}\]
Worked example
Triangle ABC has vertices A(1, 2), B(4, 3), C(2, 5). Translate the triangle by the vector (3, -2) and find the image vertices.
Step 1: The translation is (3, -2). That means 3 units to the right and 2 units down.
Step 2: The rule is (x, y) becomes (x + 3, y - 2).
Step 3: A(1, 2) becomes A'(1 + 3, 2 - 2) = A'(4, 0).
Step 4: B(4, 3) becomes B'(4 + 3, 3 - 2) = B'(7, 1).
Step 5: C(2, 5) becomes C'(2 + 3, 5 - 2) = C'(5, 3).
Check: Side AB has length the square root of ((4-1)^2 + (3-2)^2) = the square root of 10. Side A'B' has length the square root of ((7-4)^2 + (1-0)^2) = the square root of 10. They match.
The new vertices are A'(4, 0), B'(7, 1), and C'(5, 3).
Tips and common mistakes
- A translation keeps the same size, shape, and direction. The new shape is congruent to the original.
- A negative a means move left. A negative b means move down.
- Use the same numbers on every point. Every point moves by the same amount.
- A translation only slides. It does not turn or flip the shape.
- Two translations in a row, (a, b) and then (c, d), are the same as one translation of (a + c, b + d).
- The distance from each point to its image is the same for every point.
- Use primes (A', B', C') for the new points so you do not mix them up with the originals.
- Sometimes a translation is written as T(3, -2). It means the same thing as 'translate by (3, -2)'.