Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Rotation.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Math.
Concept Recap
A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.
Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A rotation moves every point around the same center by the same angle.
Common stuck point:The procedure for rotation is the easy part; the trap is rotating around the wrong center. Asking "What is the center, angle, and direction of the turn?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: What is the center, angle, and direction of the turn?
Worked Examples
Example 1
easy
Rotate the point A(3,2) by 90° counterclockwise about the origin. Where does A end up?Rotate A(3, 2) by 90° CCW about the origin
Answer
A′=(−2,3)
First step
1
Step 1: The rule for 90° CCW rotation is (x,y)→(−y,x).
Full solution
2
Step 2: Apply to A(3,2): A′=(−2,3).
3
Step 3: Verify distance from origin is unchanged: 32+22=13 and (−2)2+32=13. ✓
The 90° CCW rotation rule (x,y)→(−y,x) is derived from the rotation matrix with θ=90°: cos90°=0, sin90°=1, giving (−y,x). Rotation preserves distance from the center of rotation.
Example 2
medium
Rotate the point B(4,0) by 60° counterclockwise about the origin. Give exact coordinates.
Example 3
easy
Rotate the segment from A(1,2) to B(4,2) by 180° about the origin. Find A′B′.Segment AB rotated 180° about the origin
Example 4
medium
Rotate the point P(5,2) by 90° CCW about the center C(1,1).Rotate P(5, 2) by 90° CCW about C(1, 1)
Example 5
medium
Rotate the point (2,0) by 30° counterclockwise about the origin. Give exact coordinates.
Example 6
medium
Triangle A(2,1), B(5,1), C(5,3) is rotated 90° CCW about the origin. Find the image vertices.Triangle ABC rotated 90° CCW about the origin
Example 7
medium
Rotate the segment from A(0,0) to B(3,4) by 90° CCW about A. Find B′.Rotate segment AB by 90° CCW about A(0, 0)
Example 8
hard
Find the center of rotation for the rotation that sends A(1,0) to A′(0,1) and B(2,0) to B′(0,2).Find the center of rotation that sends A→A' and B→B'
Example 9
hard
Rotate the point (1,0) by 45° counterclockwise about the origin. Give exact coordinates.
Example 10
hard
The point (6,2) is rotated 90° CW about an unknown center and lands at (2,−2). Find the center.P(6, 2) rotated 90° CW lands at P'(2, −2) — find the center C
Example 11
challenge
Prove that the composition of two rotations by α and β about (possibly different) centers C1 and C2 is either a rotation by α+β or a translation. State precisely when each case occurs.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
What is the image of point (0,5) after a 180° rotation about the origin?Rotate (0, 5) by 180° about the origin
Example 2
hard
Triangle with vertices A(1,0), B(3,0), C(2,2) is rotated 90° CCW about the point (1,0). Find the new vertices.Triangle rotated 90° CCW about A(1, 0)
Example 3
easy
Rotate (1,0) by 90∘ counterclockwise about the origin.Rotate (1, 0) by 90° CCW about the origin
Example 4
easy
Rotate (2,3) by 180∘ about the origin.Rotate (2, 3) by 180° about the origin
Example 5
easy
Does a rotation change a figure's size?
Example 6
easy
What is the center of rotation for the rule (x,y)→(−y,x)?
Example 7
easy
Rotate (0,5) by 90∘ clockwise about the origin.Rotate (0, 5) by 90° clockwise about the origin
Example 8
easy
After rotating a figure 360∘, where does it end up?
Example 9
easy
Does the center of rotation move when you rotate a figure?
Example 10
easy
A 90∘ clockwise rotation is the same as how many degrees counterclockwise?
Example 11
medium
Rotate the point (4,2) by 270∘ counterclockwise about the origin.Rotate (4, 2) by 270° CCW about the origin
Example 12
medium
Why does rotating about a different center give a different image, even for the same angle?
Example 13
medium
Triangle vertices (1,0), (2,0), (1,3) are rotated 180∘ about the origin. Find the images.Triangle rotated 180° about the origin
Example 14
medium
A figure has rotational symmetry of order 4. What is the smallest rotation that maps it onto itself?
Example 15
medium
A rotation of 90∘ is applied twice. What single rotation is equivalent?
Example 16
medium
Does a rotation preserve orientation (clockwise/counterclockwise order of vertices)?
Example 17
medium
A point is 5 units from the center of rotation. After any rotation, how far is its image from the center?
Example 18
medium
What rotation undoes a 90∘ counterclockwise rotation?
Example 19
challenge
Rotate the point (5,2) by 90∘ counterclockwise about the center (1,1) (not the origin).Rotate P(5, 2) by 90° CCW about center C(1, 1)
Example 20
challenge
Two reflections across lines through the origin at 30∘ and 50∘ are applied. Show the result is a rotation and find its angle.
Example 21
challenge
A regular hexagon is rotated about its center. List all rotation angles (between 0∘ and 360∘) that map it onto itself.
Example 22
challenge
Explain why a rotation (other than 0∘ or 360∘) has exactly one fixed point.
Example 23
easy
Rotate the point (4,1) by 90° counterclockwise about the origin.Rotate (4, 1) by 90° CCW about the origin
Example 24
easy
Rotate the point (−3,5) by 180° about the origin.Rotate (-3, 5) by 180° about the origin
Example 25
easy
Rotate (6,0) by 270° counterclockwise about the origin.Rotate (6, 0) by 270° CCW about the origin
Example 26
easy
Rotate the point (2,−7) by 90° clockwise about the origin.
Example 27
medium
Rotate the point (3,5) by 180° about the center (2,1).Rotate (3, 5) by 180° about center (2, 1)
Example 28
medium
A figure is rotated 90° CCW, then 90° CCW again about the same center. What single rotation has the same effect?
Example 29
medium
Point (4,6) is rotated about the origin and lands at (−6,4). What was the rotation angle?What rotation maps P(4, 6) to P'(−6, 4)?
Example 30
medium
A regular hexagon has a center of rotational symmetry. What is the smallest angle of rotation (in degrees) that maps the hexagon to itself?
Example 31
medium
After rotating the point (7,−2) by 180° about the center (3,4), where does it land?Rotate P(7, −2) by 180° about center C(3, 4)
Example 32
hard
The point (5,3) is rotated 90° CCW about an unknown center (a,b) and lands at (3,7). Find (a,b).
Example 33
hard
Triangle with vertices A(0,0), B(4,0), C(4,3) is rotated 90° CCW about B. Find the new coordinates of A and C.Triangle ABC rotated 90° CCW about vertex B(4, 0)
Example 34
hard
What single rotation about the origin equals the composition of a 250° CCW rotation followed by a 190° CCW rotation about the origin?