Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A translation moves a figure without turning, flipping, or resizing it.
Common stuck point:The procedure for translation is the easy part; the trap is changing only one coordinate when the move has horizontal and vertical parts. Asking "Did every point move by the same vector?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Did every point move by the same vector?
Worked Examples
Example 1
easy
Translate the point A(2,−3) by the vector (5,4). Where does A end up?Translate A(2, −3) by the vector (5, 4)
Answer
A′=(7,1)
First step
1
Step 1: A translation by (a,b) maps (x,y)→(x+a,y+b).
Full solution
2
Step 2: A(2,−3) translated by (5,4): (2+5,−3+4)=(7,1).
3
Step 3: A′=(7,1).
A translation slides every point by the same amount in the same direction. The translation vector (5,4) means move 5 units right and 4 units up. Translations preserve distances, angles, and orientation — they are rigid motions.
Example 2
medium
Triangle with vertices A(0,0), B(3,0), C(0,4) is translated by (−2,3). Find the new vertices and confirm the side lengths are unchanged.Triangle ABC translated by (−2, 3)
Example 3
medium
Triangle with vertices A(1,2), B(4,2), C(3,5) is translated by ⟨−3,4⟩. Find the new vertices.Triangle ABC translated by ⟨−3, 4⟩
Example 4
medium
Quadrilateral WXYZ has vertices W(1,1),X(4,1),Y(5,3),Z(2,3). Translate by ⟨−5,2⟩ and list the image vertices.Quadrilateral WXYZ translated by ⟨−5, 2⟩
Example 5
medium
A figure with vertex A(5,2) is translated so A→A′(−1,6). Where does the vertex B(3,−4) go under the same translation?Same translation maps A → A′; find where B goes
Example 6
medium
Translate the parabola y=x2 by ⟨3,−2⟩. Write the equation of the image.
Example 7
medium
Triangle ABC with A(0,0),B(6,0),C(3,4) is translated so C goes to the origin. Find the images of A and B.Translate triangle ABC so that C goes to the origin
Example 8
hard
Vector u=⟨3,4⟩. A second translation has magnitude 13 and is parallel to u. Write its vector form (assuming same direction).
Example 9
hard
Show that a translation maps a line to a parallel line. Take line y=−x+4 and translation ⟨2,1⟩ as an example.
Example 10
hard
A robot at (0,0) executes the moves: ⟨2,3⟩, ⟨−1,4⟩, ⟨5,−2⟩. Where is the robot?Robot executes three moves — trace the path to find final position
Example 11
challenge
Prove that the composition of two translations is itself a translation, and find the vector.
Example 12
challenge
Suppose translation T1 takes (0,0)→(3,4) and translation T2 takes (3,4)→(−1,7). Find the single translation equivalent to T2∘T1 and the total distance traveled by the origin.T₁ takes O→P, T₂ takes P→Q — find the combined single translation
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Point P(4,7) is translated to P′(1,3). What is the translation vector?P maps to P′ — find the translation vector
Example 2
hard
A square with vertices (1,1), (3,1), (3,3), (1,3) is translated so that the bottom-left corner moves to (0,0). Find the new vertices and the translation vector.Square slides until its bottom-left corner reaches the origin
Example 3
easy
Translate the point (2,5) by 3 right and 2 up.Translate (2, 5) by 3 right and 2 up
Example 4
easy
Translate (4,1) by 2 left.
Example 5
easy
Does a translation change the size or shape of a figure?
Example 6
easy
A translation rule is (x,y)→(x−4,y+1). Which way does it move points?
Example 7
easy
Translate the point (0,0) by the vector ⟨5,−3⟩.
Example 8
easy
After translating a triangle, how do the image's side lengths compare to the original?
Example 9
easy
Translate (3,3) by 0 right and 4 down.
Example 10
easy
Does a translation preserve the orientation (clockwise/counterclockwise order) of a figure?
Example 11
medium
Triangle has vertices (1,1), (4,1), (1,5). Translate by ⟨2,3⟩. Find the new vertices.Triangle translated by ⟨2, 3⟩
Example 12
medium
A point (2,7) is translated to (6,4). Find the translation vector.(2, 7) is translated to (6, 4) — find the vector
Example 13
medium
A figure is translated by ⟨3,2⟩, then by ⟨−1,4⟩. Find the single equivalent translation.Two successive translations — what single vector has the same net effect?
Example 14
medium
Why does a translation never have a fixed point (a point that maps to itself), unless the shift is zero?
Example 15
medium
A square is translated. Are its sides still parallel to their original positions?
Example 16
medium
A translation maps A(1,2)→A′(4,6). Using the same translation, find the image of B(0,0).Same translation maps A → A′ and B → B′
Example 17
medium
A figure is translated by ⟨a,b⟩. What single translation undoes it?
Example 18
medium
Does a translation change a figure's area?
Example 19
challenge
Two reflections across parallel vertical lines x=1 and x=4 are applied in turn. Show the result is a translation and find its vector.
Example 20
challenge
A repeating wallpaper pattern looks identical after sliding it 5 cm right. What is this property called, and what does it imply about sliding it 10 cm right?
Example 21
challenge
A point is translated by ⟨3,4⟩. How far does it move, and does this distance depend on which point you start from?
Example 22
challenge
Explain why translating a line always produces a line parallel to (or identical to) the original.
Example 23
easy
Translate the point (6,2) by the vector ⟨−4,5⟩.
Example 24
easy
A translation maps (2,3)→(8,7). What is the translation vector?(2, 3) maps to (8, 7) — what is the translation vector?
Example 25
easy
Translate the point (−3,−5) by the rule (x,y)→(x+7,y+2).
Example 26
easy
A boat at (2,1) sails 5 units east and 3 units south. Where is the boat now?Boat sails 5 east then 3 south — find final position
Example 27
medium
After translating by ⟨3,−4⟩, then by ⟨−1,6⟩, what single translation vector has the same effect?
Example 28
medium
A triangle is translated by ⟨a,b⟩. If a=4 and b=−3, what is the distance each vertex moves?
Example 29
medium
Translate line y=2x+1 by the vector ⟨0,4⟩. Write the equation of the image.Line y = 2x + 1 translated by ⟨0, 4⟩ becomes y = 2x + 5
Example 30
medium
After two translations of a point, the net effect is ⟨7,3⟩. If the first was ⟨4,−2⟩, what was the second?
Example 31
medium
A circle of radius 3 centered at (2,−1) is translated by ⟨4,6⟩. What is the center of the image circle, and what is its radius?Circle of radius 3 at (2, −1) translates to center (6, 5)
Example 32
hard
A point is reflected over the x-axis, then translated by ⟨2,3⟩. If the final position is (5,1), what was the original point?
Example 33
hard
Triangle T1 has vertices (1,1),(4,1),(4,5). Triangle T2 is congruent to T1 and is obtained by a translation. If one vertex of T2 is (6,3) and it corresponds to (4,1) in T1, what translation vector was used?T₂ is T₁ translated — vertex (4,1) goes to (6,3)
Example 34
hard
A translation maps the circle (x−1)2+(y+2)2=9 to a circle centered at (7,4). What is the translation vector?Circle centered at (1, −2) translates to center (7, 4) — find the vector
Example 35
hard
A figure is translated by ⟨4,−1⟩, then by ⟨−4,1⟩. Describe the net effect.