Practice Translation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A rigid transformation that slides every point of a figure the same distance in the same direction.

Sliding a chess piece straight across the board—every point moves the same amount, same direction.

Showing a random 20 of 50 problems.

Example 1

hard
A translation maps the circle (x−1)2+(y+2)2=9(x-1)^2 + (y+2)^2 = 9 to a circle centered at (7,4)(7, 4). What is the translation vector?

Example 2

medium
After translating by ⟨3,−4⟩\langle 3, -4 \rangle, then by ⟨−1,6⟩\langle -1, 6 \rangle, what single translation vector has the same effect?

Example 3

easy
Point P(4,7)P(4, 7) is translated to P′(1,3)P'(1, 3). What is the translation vector?

Example 4

medium
A figure with vertex A(5,2)A(5, 2) is translated so A→A′(−1,6)A \to A'(-1, 6). Where does the vertex B(3,−4)B(3, -4) go under the same translation?

Example 5

medium
Does a translation change a figure's area?

Example 6

easy
What is the translation vector that leaves every point fixed?

Example 7

medium
Translate the parabola y=x2y = x^2 by ⟨3,−2⟩\langle 3, -2 \rangle. Write the equation of the image.

Example 8

easy
Does a translation change the size or shape of a figure?

Example 9

hard
A square with vertices (1,1)(1,1), (3,1)(3,1), (3,3)(3,3), (1,3)(1,3) is translated so that the bottom-left corner moves to (0,0)(0, 0). Find the new vertices and the translation vector.

Example 10

challenge
A repeating wallpaper pattern looks identical after sliding it 55 cm right. What is this property called, and what does it imply about sliding it 1010 cm right?

Example 11

medium
A figure is translated by ⟨a,b⟩\langle a, b \rangle. What single translation undoes it?

Example 12

challenge
Suppose translation T1T_1 takes (0,0)→(3,4)(0,0) \to (3, 4) and translation T2T_2 takes (3,4)→(−1,7)(3, 4) \to (-1, 7). Find the single translation equivalent to T2∘T1T_2 \circ T_1 and the total distance traveled by the origin.

Example 13

medium
Triangle has vertices (1,1)(1,1), (4,1)(4,1), (1,5)(1,5). Translate by ⟨2,3⟩\langle 2, 3 \rangle. Find the new vertices.

Example 14

medium
Translation is an isometry. What does isometry mean?

Example 15

medium
Why does a translation never have a fixed point (a point that maps to itself), unless the shift is zero?

Example 16

medium
A circle of radius 33 centered at (2,−1)(2, -1) is translated by ⟨4,6⟩\langle 4, 6 \rangle. What is the center of the image circle, and what is its radius?

Example 17

medium
A figure is translated by ⟨3,2⟩\langle 3, 2 \rangle, then by ⟨−1,4⟩\langle -1, 4 \rangle. Find the single equivalent translation.

Example 18

easy
Translate the point (2,5)(2, 5) by 3 right and 2 up.

Example 19

challenge
Prove that the composition of two translations is itself a translation, and find the vector.

Example 20

easy
A translation rule is (x,y)→(x−4,y+1)(x, y) \to (x - 4, y + 1). Which way does it move points?
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