What Are Translations?

In geometry, a translation is a type of transformation that slides a shape from one position to another without rotating, resizing, or flipping it. Every point of the shape moves the same distance in the same direction. A translation is described by a vector that indicates the direction and distance of the movement.

Translation Vector

A translation is represented by a vector of the form:

$$\binom{a}{b}$$
Where:

• $$a$$ represents the horizontal movement (left or right).
• $$b$$ represents the vertical movement (up or down).

For example:

• The vector $$\binom{3}{2}$$ means the shape moves 3 units to the right and 2 units up.
• The vector $$\binom{-2}{-3}$$ means the shape moves 2 units to the left and 3 units down.

Finding Coordinates After a Translation

When you translate a point or shape, you adjust the coordinates of each point by the values given in the vector.

For a point with coordinates

$$(x, y)$$ :

• After a translation by vector $$\binom{a}{b}$$ , the new coordinates are:
  $$(x’, y’) = (x + a, y + b)$$

For example, if the point

$$(2, 3)$$ is translated by the vector $$\binom{4}{-2}$$ , the new coordinates will be:
$$(x’, y’) = (2 + 4, 3 – 2) = (6, 1)$$

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Practice Questions: Translations and Finding Coordinates

Easy Level Questions

1. Translate the point (2, 3) by the vector $$\binom{1}{1}$$ .
2. What are the new coordinates of the point (-1, 4) after translating by $$\binom{3}{0}$$ ?
3. Translate the point (5, 2) by the vector $$\binom{-2}{3}$$ .
4. Find the coordinates of the point (0, 0) after translating by $$\binom{4}{5}$$ .
5. Translate the point (-3, 1) by the vector $$\binom{2}{-1}$$ .
6. What are the new coordinates of the point (6, 7) after translating by $$\binom{0}{-4}$$ ?
7. Translate the point (4, 0) by the vector $$\binom{3}{2}$$ .
8. Find the coordinates of the point (1, -1) after translating by $$\binom{2}{3}$$ .
9. Translate the point (-2, 3) by the vector $$\binom{-1}{2}$$ .
10. What are the new coordinates of the point (7, 4) after translating by $$\binom{0}{-3}$$ ?
11. Translate the point (0, 5) by the vector $$\binom{1}{-1}$$ .
12. What are the new coordinates of the point (3, 0) after translating by $$\binom{4}{4}$$ ?
13. Translate the point (-5, -2) by the vector $$\binom{3}{1}$$ .
14. Find the new coordinates of the point (2, 6) after translating by $$\binom{-1}{-2}$$ .
15. What are the new coordinates of the point (-3, 4) after translating by $$\binom{0}{3}$$ ?
16. Translate the point (5, -1) by the vector $$\binom{-2}{1}$$ .
17. What are the new coordinates of the point (1, 1) after translating by $$\binom{5}{-2}$$ ?
18. Translate the point (-4, 3) by the vector $$\binom{1}{-1}$$ .
19. Find the new coordinates of the point (0, -3) after translating by $$\binom{2}{5}$$ .
20. What are the new coordinates of the point (3, -2) after translating by $$\binom{-3}{2}$$ ?

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Medium Level Questions

1. Translate the point (6, -4) by the vector $$\binom{-3}{1}$$ .
2. What are the new coordinates of the point (-5, 7) after translating by $$\binom{2}{-3}$$ ?
3. Translate the point (2, 5) by the vector $$\binom{-1}{4}$$ .
4. Find the coordinates of the point (0, -1) after translating by $$\binom{5}{6}$$ .
5. Translate the point (-3, -3) by the vector $$\binom{4}{-2}$$ .
6. What are the new coordinates of the point (4, 8) after translating by $$\binom{-2}{3}$$ ?
7. Translate the point (5, 2) by the vector $$\binom{3}{-5}$$ .
8. Find the new coordinates of the point (-6, 4) after translating by $$\binom{2}{1}$$ .
9. What are the new coordinates of the point (7, -2) after translating by $$\binom{-5}{4}$$ ?
10. Translate the point (-1, -5) by the vector $$\binom{6}{2}$$ .
11. Find the new coordinates of the point (0, 3) after translating by $$\binom{-2}{-1}$$ .
12. Translate the point (-4, -6) by the vector $$\binom{1}{3}$$ .
13. What are the new coordinates of the point (3, -3) after translating by $$\binom{-2}{4}$$ ?
14. Translate the point (-7, 1) by the vector $$\binom{5}{-1}$$ .
15. What are the new coordinates of the point (0, 0) after translating by $$\binom{-4}{-6}$$ ?
16. Translate the point (8, -3) by the vector $$\binom{-3}{2}$$ .
17. What are the new coordinates of the point (-2, -4) after translating by $$\binom{1}{5}$$ ?
18. Translate the point (7, 7) by the vector $$\binom{-5}{-7}$$ .
19. Find the new coordinates of the point (0, 9) after translating by $$\binom{3}{-5}$$ .
20. What are the new coordinates of the point (-5, -1) after translating by $$\binom{4}{6}$$ ?

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Hard Level Questions

1. Translate the point (10, -6) by the vector $$\binom{-5}{3}$$ .
2. What are the new coordinates of the point (-12, 8) after translating by $$\binom{6}{-7}$$ ?
3. Translate the point (0, 12) by the vector $$\binom{4}{-9}$$ .
4. Find the new coordinates of the point (-5, 10) after translating by $$\binom{-3}{-5}$$ .
5. Translate the point (9, -3) by the vector $$\binom{-2}{6}$$ .
6. What are the new coordinates of the point (-7, -7) after translating by $$\binom{8}{-1}$$ ?
7. Translate the point (3, -11) by the vector $$\binom{-4}{8}$$ .
8. What are the new coordinates of the point (5, -6) after translating by $$\binom{-7}{4}$$ ?
9. Find the new coordinates of the point (-8, 3) after translating by $$\binom{5}{-9}$$ .
10. Translate the point (12, -8) by the vector $$\binom{-10}{3}$$ .
11. What are the new coordinates of the point (-3, -2) after translating by $$\binom{6}{8}$$ ?
12. Translate the point (8, 12) by the vector $$\binom{-7}{-4}$$ .
13. What are the new coordinates of the point (-9, -12) after translating by $$\binom{3}{7}$$ ?
14. Translate the point (-5, 0) by the vector $$\binom{6}{-2}$$ .
15. What are the new coordinates of the point (0, -7) after translating by $$\binom{-8}{5}$$ ?
16. Translate the point (-11, -3) by the vector $$\binom{5}{4}$$ .
17. What are the new coordinates of the point (6, -6) after translating by $$\binom{-9}{2}$$ ?
18. Translate the point (-6, 5) by the vector $$\binom{3}{-8}$$ .
19. Find the new coordinates of the point (13, 0) after translating by $$\binom{-5}{-9}$$ .
20. What are the new coordinates of the point (7, -4) after translating by $$\binom{-6}{7}$$ ?

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Answers with Explanation

Easy Level Answers

1. Answer: $$(3, 4)$$
   Explanation: Add 1 to the x-coordinate and 1 to the y-coordinate:
   $$(2 + 1, 3 + 1) = (3, 4)$$
2. Answer: $$(2, 4)$$
   Explanation: Add 3 to the x-coordinate (no change to y-coordinate):
   $$(-1 + 3, 4 + 0) = (2, 4)$$
3. Answer: $$(3, 5)$$
   Explanation: Add -2 to the x-coordinate and 3 to the y-coordinate:
   $$(5 – 2, 2 + 3) = (3, 5)$$
4. Answer: $$(4, 5)$$
   Explanation: Add 4 to the x-coordinate and 5 to the y-coordinate:
   $$(0 + 4, 0 + 5) = (4, 5)$$
5. Answer: $$(-1, 0)$$
   Explanation: Add 2 to the x-coordinate and -1 to the y-coordinate:
   $$(-3 + 2, 1 – 1) = (-1, 0)$$
6. Answer: $$(6, 3)$$
   Explanation: Add 0 to the x-coordinate and -4 to the y-coordinate:
   $$(6 + 0, 7 – 4) = (6, 3)$$
7. Answer: $$(7, 2)$$
   Explanation: Add 3 to the x-coordinate and 2 to the y-coordinate:
   $$(4 + 3, 0 + 2) = (7, 2)$$
8. Answer: $$(3, 2)$$
   Explanation: Add 2 to the x-coordinate and 3 to the y-coordinate:
   $$(1 + 2, -1 + 3) = (3, 2)$$
9. Answer: $$(-3, 5)$$
   Explanation: Add -1 to the x-coordinate and 2 to the y-coordinate:
   $$(-2 – 1, 3 + 2) = (-3, 5)$$
10. Answer: $$(7, 1)$$
    Explanation: Add 0 to the x-coordinate and -3 to the y-coordinate:
    $$(7 + 0, 4 – 3) = (7, 1)$$
11. Answer: $$(1, 4)$$
    Explanation: Add 1 to the x-coordinate and -1 to the y-coordinate:
    $$(0 + 1, 5 – 1) = (1, 4)$$
12. Answer: $$(7, 4)$$
    Explanation: Add 4 to both the x- and y-coordinates:
    $$(3 + 4, 0 + 4) = (7, 4)$$
13. Answer: $$(-2, -1)$$
    Explanation: Add 3 to the x-coordinate and 1 to the y-coordinate:
    $$(-5 + 3, -2 + 1) = (-2, -1)$$
14. Answer: $$(1, 4)$$
    Explanation: Add -1 to the x-coordinate and -2 to the y-coordinate:
    $$(2 – 1, 6 – 2) = (1, 4)$$
15. Answer: $$(-3, 7)$$
    Explanation: Add 0 to the x-coordinate and 3 to the y-coordinate:
    $$(-3 + 0, 4 + 3) = (-3, 7)$$
16. Answer: $$(3, 0)$$
    Explanation: Add -2 to the x-coordinate and 1 to the y-coordinate:
    $$(5 – 2, -1 + 1) = (3, 0)$$
17. Answer: $$(6, -1)$$
    Explanation: Add 5 to the x-coordinate and -2 to the y-coordinate:
    $$(1 + 5, 1 – 2) = (6, -1)$$
18. Answer: $$(-3, 2)$$
    Explanation: Add 1 to the x-coordinate and -1 to the y-coordinate:
    $$(-4 + 1, 3 – 1) = (-3, 2)$$
19. Answer: $$(2, 2)$$
    Explanation: Add 2 to the x-coordinate and 5 to the y-coordinate:
    $$(0 + 2, -3 + 5) = (2, 2)$$
20. Answer: $$(0, 0)$$
    Explanation: Add -3 to the x-coordinate and 2 to the y-coordinate:
    $$(3 – 3, -2 + 2) = (0, 0)$$

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Medium Level Answers

1. Answer: $$(3, -3)$$
   Explanation: Add -3 to the x-coordinate and 1 to the y-coordinate:
   $$(6 – 3, -4 + 1) = (3, -3)$$
2. Answer: $$(-3, 4)$$
   Explanation: Add 2 to the x-coordinate and -3 to the y-coordinate:
   $$(-5 + 2, 7 – 3) = (-3, 4)$$
3. Answer: $$(1, 9)$$
   Explanation: Add -1 to the x-coordinate and 4 to the y-coordinate:
   $$(2 – 1, 5 + 4) = (1, 9)$$
4. Answer: $$(5, 5)$$
   Explanation: Add 5 to the x-coordinate and 6 to the y-coordinate:
   $$(0 + 5, -1 + 6) = (5, 5)$$
5. Answer: $$(1, -5)$$
   Explanation: Add 4 to the x-coordinate and -2 to the y-coordinate:
   $$(-3 + 4, -3 – 2) = (1, -5)$$
6. Answer: $$(2, 11)$$
   Explanation: Add -2 to the x-coordinate and 3 to the y-coordinate:
   $$(4 – 2, 8 + 3) = (2, 11)$$
7. Answer: $$(8, -3)$$
   Explanation: Add 3 to the x-coordinate and -5 to the y-coordinate:
   $$(5 + 3, 2 – 5) = (8, -3)$$
8. Answer: $$(-4, 5)$$
   Explanation: Add 2 to the x-coordinate and 1 to the y-coordinate:
   $$(-6 + 2, 4 + 1) = (-4, 5)$$
9. Answer: $$(2, 2)$$
   Explanation: Add -5 to the x-coordinate and 4 to the y-coordinate:
   $$(7 – 5, -2 + 4) = (2, 2)$$
10. Answer: $$(5, -3)$$
    Explanation: Add 6 to the x-coordinate and 2 to the y-coordinate:
    $$(-1 + 6, -5 + 2) = (5, -3)$$
11. Answer: $$(-2, 2)$$
    Explanation: Add -2 to the x-coordinate and -1 to the y-coordinate:
    $$(0 – 2, 3 – 1) = (-2, 2)$$
12. Answer: $$(-3, -3)$$
    Explanation: Add 1 to the x-coordinate and 3 to the y-coordinate:
    $$(-4 + 1, -6 + 3) = (-3, -3)$$
13. Answer: $$(1, 1)$$
    Explanation: Add -2 to the x-coordinate and 4 to the y-coordinate:
    $$(3 – 2, -3 + 4) = (1, 1)$$
14. Answer: $$(-2, 0)$$
    Explanation: Add 5 to the x-coordinate and -1 to the y-coordinate:
    $$(-7 + 5, 1 – 1) = (-2, 0)$$
15. Answer: $$(-4, -6)$$
    Explanation: Add -4 to the x-coordinate and -6 to the y-coordinate:
    $$(0 – 4, 0 – 6) = (-4, -6)$$
16. Answer: $$(5, -1)$$
    Explanation: Add -3 to the x-coordinate and 2 to the y-coordinate:
    $$(8 – 3, -3 + 2) = (5, -1)$$
17. Answer: $$(-1, 1)$$
    Explanation: Add 1 to the x-coordinate and 5 to the y-coordinate:
    $$(-2 + 1, -4 + 5) = (-1, 1)$$
18. Answer: $$(2, 0)$$
    Explanation: Add -5 to the x-coordinate and -7 to the y-coordinate:
    $$(7 – 5, 7 – 7) = (2, 0)$$
19. Answer: $$(3, 4)$$
    Explanation: Add 3 to the x-coordinate and -5 to the y-coordinate:
    $$(0 + 3, 9 – 5) = (3, 4)$$
20. Answer: $$(-1, 5)$$
    Explanation: Add 4 to the x-coordinate and 6 to the y-coordinate:
    $$(-5 + 4, -1 + 6) = (-1, 5)$$

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Hard Level Answers

Answer:

$$(1, 3)$$
Explanation: Add -6 to the x-coordinate and 7 to the y-coordinate:
$$(7 – 6, -4 + 7) = (1, 3)$$

Answer:

$$(5, -3)$$
Explanation: Add -5 to the x-coordinate and 3 to the y-coordinate:
$$(10 – 5, -6 + 3) = (5, -3)$$

Answer:

$$(-6, 1)$$
Explanation: Add 6 to the x-coordinate and -7 to the y-coordinate:
$$(-12 + 6, 8 – 7) = (-6, 1)$$

Answer:

$$(4, 3)$$
Explanation: Add 4 to the x-coordinate and -9 to the y-coordinate:
$$(0 + 4, 12 – 9) = (4, 3)$$

Answer:

$$(-8, 5)$$
Explanation: Add -3 to the x-coordinate and -5 to the y-coordinate:
$$(-5 – 3, 10 – 5) = (-8, 5)$$

Answer:

$$(7, 3)$$
Explanation: Add -2 to the x-coordinate and 6 to the y-coordinate:
$$(9 – 2, -3 + 6) = (7, 3)$$

Answer:

$$(1, -8)$$
Explanation: Add 8 to the x-coordinate and -1 to the y-coordinate:
$$(-7 + 8, -7 – 1) = (1, -8)$$

Answer:

$$(-1, -3)$$
Explanation: Add -4 to the x-coordinate and 8 to the y-coordinate:
$$(3 – 4, -11 + 8) = (-1, -3)$$

Answer:

$$(-2, -2)$$
Explanation: Add -7 to the x-coordinate and 4 to the y-coordinate:
$$(5 – 7, -6 + 4) = (-2, -2)$$

Answer:

$$(-3, -6)$$
Explanation: Add 5 to the x-coordinate and -9 to the y-coordinate:
$$(-8 + 5, 3 – 9) = (-3, -6)$$

Answer:

$$(2, -5)$$
Explanation: Add -10 to the x-coordinate and 3 to the y-coordinate:
$$(12 – 10, -8 + 3) = (2, -5)$$

Answer:

$$(3, 6)$$
Explanation: Add 6 to the x-coordinate and 8 to the y-coordinate:
$$(-3 + 6, -2 + 8) = (3, 6)$$

Answer:

$$(1, 8)$$
Explanation: Add -7 to the x-coordinate and -4 to the y-coordinate:
$$(8 – 7, 12 – 4) = (1, 8)$$

Answer:

$$(-6, -5)$$
Explanation: Add 3 to the x-coordinate and 7 to the y-coordinate:
$$(-9 + 3, -12 + 7) = (-6, -5)$$

Answer:

$$(1, -2)$$
Explanation: Add 6 to the x-coordinate and -2 to the y-coordinate:
$$(-5 + 6, 0 – 2) = (1, -2)$$

Answer:

$$(-8, -2)$$
Explanation: Add -8 to the x-coordinate and 5 to the y-coordinate:
$$(0 – 8, -7 + 5) = (-8, -2)$$

Answer:

$$(-6, 1)$$
Explanation: Add 5 to the x-coordinate and 4 to the y-coordinate:
$$(-11 + 5, -3 + 4) = (-6, 1)$$

Answer:

$$(-3, -4)$$
Explanation: Add -9 to the x-coordinate and 2 to the y-coordinate:
$$(6 – 9, -6 + 2) = (-3, -4)$$

Answer:

$$(-3, -3)$$
Explanation: Add 3 to the x-coordinate and -8 to the y-coordinate:
$$(-6 + 3, 5 – 8) = (-3, -3)$$

Answer:

$$(8, -9)$$
Explanation: Add -5 to the x-coordinate and -9 to the y-coordinate:
$$(13 – 5, 0 – 9) = (8, -9)$$