Translations, Reflections & Rotations — Challenge
Three moves — slide, flip, turn — never change a shape. Everything in this unit follows from that one fact.
What these problems 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
The point (2,3) is translated 4 units right and 1 unit down. What are its new coordinates?
- (6,2)
- (-2,4)
- (6,4)
- (2,2)
- (8,-3)
Worked answer
(6,2)
Moving right adds to the x coordinate: 2+4 = 6.
Moving down subtracts from the y coordinate: 3-1 = 2.
So the image is (6,2).
Problem 2
The point (3,0) is rotated 90 degrees anticlockwise about the origin. What are its new coordinates?
- (0,-3)
- (-3,0)
- (0,3)
- (3,0)
- (-3,3)
Worked answer
(0,3)
The point sits 3 units along the positive x-axis.
Turning a quarter circle anticlockwise about the origin carries the positive x-axis onto the positive y-axis.
So the image is (0,3). A clockwise quarter turn would give (0,-3) instead.
Problem 3
The point (-2,5) is reflected in the y-axis. What are its new coordinates?
- (-2,-5)
- (2,5)
- (5,-2)
- (2,-5)
- (-2,5)
Worked answer
(2,5)
Reflecting in the y-axis flips the point left to right, so the sign of x changes and y stays put.
(-2,5) becomes (2,5).
Reflecting in the x-axis instead would give (-2,-5).
Problem 4
Which transformation moves a figure without changing its size, shape or orientation?
- Translation
- Dilation
- Stretch
- Shear
- Reflection
Worked answer
Translation
A translation slides every point the same distance in the same direction.
Nothing is turned, flipped or resized, so size, shape and orientation all survive.
A reflection also preserves size and shape but reverses orientation, so it does not qualify.
Problem 5
The point (1,2) is rotated 180 degrees about the origin. What are its new coordinates?
- (1,-2)
- (-1,2)
- (-1,-2)
- (2,1)
- (-2,-1)
Worked answer
(-1,-2)
A half turn about the origin sends every point to the opposite side, the same distance away.
Both coordinates change sign, so (1,2) becomes (-1,-2).
A half turn is the one rotation where the result does not depend on the direction of turning.
Problem 6
A figure and its image are congruent. Which transformation could not have produced the image?
- A translation
- A rotation
- A reflection
- A dilation with scale factor 2
- A rotation followed by a translation
Worked answer
A dilation with scale factor 2
Congruent means identical in size and shape.
Translations, rotations and reflections all preserve lengths, and so does any combination of them.
A dilation with scale factor 2 doubles every length, so the image is similar but larger, not congruent.
Problem 7
The point (4,-3) is reflected in the x-axis. What are its new coordinates?
- (-4,-3)
- (4,3)
- (-4,3)
- (3,4)
- (4,-3)
Worked answer
(4,3)
Reflecting in the x-axis flips the point above or below that line, so the sign of y changes.
(4,-3) becomes (4,3).
Problem 8
A shape is translated 3 units right and 2 units down. A point (x,y) on it moves to which position?
- (x+3,y-2)
- (x-3,y+2)
- (x+3,y+2)
- (x-2,y+3)
- (x+2,y-3)
Worked answer
(x+3,y-2)
Right adds to x, so x becomes x+3.
Down subtracts from y, so y becomes y-2.
The rule is therefore (x,y)→(x+3,y-2), and it applies to every point of the shape alike.