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:
$$
(x’, y’) = (2 + 4, 3 – 2) = (6, 1)
$$

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 $$ \binom{-3}{2} $$?

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 $$ \binom{4}{6} $$?

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 $$ \binom{-6}{7} $$?

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:
    $$ (3 – 3, -2 + 2) = (0, 0) $$

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:
    $$ (-5 + 4, -1 + 6) = (-1, 5) $$

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:
$$ (13 – 5, 0 – 9) = (8, -9) $$