Translations, Reflections & Rotations — Practice
Three moves — slide, flip, turn — never change a shape. Everything in this unit follows from that one fact.
What these questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Image — The shape after the move. Its corners get primes: A', B', C'.
- Rigid motion — A move that keeps every length and every angle. Slides, flips and turns are all rigid.
- Translation — A slide. Every point moves the same distance in the same direction.
- Reflection — A flip across a line, like a mirror.
- Rotation — A turn about a fixed point, through a given angle and direction.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Transformation — A change to the function that moves or stretches its graph.
Problem 1
Translate A(3, 5) by the rule (x, y) → (x + 4, y - 2).
Worked answer
(7, 3)
3 + 4 = 7
5 - 2 = 3
A'(7, 3)
Problem 2
Translate B(-2, 7) five units left and three units down.
Worked answer
(-7, 4)
-2 - 5 = -7
7 - 3 = 4
B'(-7, 4)
Problem 3
Reflect P(6, -4) across the x-axis.
Worked answer
(6, -(-4)) = (6, 4)
x stays 6
-4 becomes 4
P'(6, 4)
Problem 4
Reflect Q(-3, 8) across the y-axis.
Worked answer
(3, 8)
-3 becomes 3
y stays 8
Q'(3, 8)
Problem 5
Rotate R(4, 1) by 180° about the origin.
Worked answer
(-4, -1)
Rule: (x, y) → (-x, -y)
R'(-4, -1)
Problem 6
Rotate S(2, 7) by 90° counter-clockwise about the origin.
Worked answer
(-7, 2)
Swap: (7, 2)
Negate the front: (-7, 2)
S'(-7, 2)
Problem 7
Rotate T(-5, 3) by 270° counter-clockwise about the origin.
Worked answer
(3, 5)
y = 3 moves to the front
-x = 5
T'(3, 5)
Problem 8
A triangle has vertices (1, 1), (4, 1), (4, 5). After a translation, the image of (1, 1) is (-2, 6). Where does (4, 5) go?
Worked answer
(1, 10)
1 → -2 means -3 on x
1 → 6 means +5 on y
(4 - 3, 5 + 5) = (1, 10)
Problem 9
Point (5, -2) maps to (-5, -2). Which single transformation was used?
Worked answer
reflection across the y-axis
The x changed sign; the y did not
That is the rule (x, y) → (-x, y)
A reflection across the y-axis
Problem 10
A figure is moved so that every point travels 6 units right and 0 units vertically. Which is true of the image?
- It is congruent to the original
- It is larger than the original
- It is a mirror image of the original
- It has different angle measures
Worked answer
It is congruent to the original
A translation changes position only
Lengths and angles are unchanged
So the image is congruent
Problem 11
Reflect (a, b) across the x-axis, then across the y-axis. Give the result.
Worked answer
(-a, -b)
Across the x-axis: (a, -b)
Across the y-axis: (-a, -b)
Same as a 180° rotation
Problem 12
Point M(2, -3) is rotated 90° counter-clockwise about the origin, then translated 1 unit up. Find the image.
Worked answer
(3, 3)
Rotate: (2, -3) → (3, 2)
Translate up 1: (3, 3)
Problem 13
A point is reflected across the y-axis and then translated 4 right. The final image is (1, 6). Find the original point.
Worked answer
(3, 6)
Undo the translation: (1 - 4, 6) = (-3, 6)
Undo the reflection: (3, 6)
Check: (3,6) → (-3,6) → (1,6) ✓
Problem 14
How many degrees counter-clockwise about the origin will map (4, 0) to (0, -4)?
Worked answer
270
90° takes it to (0, 4)
180° to (-4, 0)
270° to (0, -4)
Problem 15
Triangle ABC has A(1, 2), B(5, 2), C(1, 6). Its image is A'(2, -1), B'(2, -5), C'(6, -1). Which transformation was used?
Worked answer
rotation of 90 degrees clockwise about the origin
A(1,2) → (2, -1) ✓
B(5,2) → (2, -5) ✓
C(1,6) → (6, -1) ✓
That rule is a 90° clockwise rotation, also called 270° counter-clockwise
Problem 16
A translation of 3 right and 2 up is followed by another of 1 left and 5 down. Write the single rule that does both.
Worked answer
(x + 2, y - 3)
Horizontal: +3 - 1 = +2
Vertical: +2 - 5 = -3
(x, y) → (x + 2, y - 3)