Rotations: Step-by-Step Guide
A rotation turns a shape around a fixed point (the center) by a certain angle. When the center is the origin, there are simple rules. 90 degrees counterclockwise: (x, y) becomes (-y, x). 180 degrees: (x, y) becomes (-x, -y). 270 degrees counterclockwise: (x, y) becomes (y, -x). A rotation keeps the same size and shape.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Find the center and the angle. Find the center of rotation (usually the origin), the angle, and the direction. Counterclockwise is the positive direction.\[\text{center} = (0, 0),\; \theta = 90^\circ,\;180^\circ,\;270^\circ\]
- Use the rotation rule (center at the origin). 90 degrees counterclockwise: (x, y) becomes (-y, x). 180 degrees: (x, y) becomes (-x, -y). 270 degrees counterclockwise (same as 90 degrees clockwise): (x, y) becomes (y, -x).\[\begin{aligned}90^\circ &: (x, y) \mapsto (-y, x)\\180^\circ &: (x, y) \mapsto (-x, -y)\\270^\circ &: (x, y) \mapsto (y, -x)\end{aligned}\]
- A rule for any angle (advanced). For any angle, a point (x, y) rotates to (x cos(angle) - y sin(angle), x sin(angle) + y cos(angle)). You will use this more in later classes.\[\begin{pmatrix}x'\\y'\end{pmatrix} = \begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\]
- Rotate each vertex. Use the rule on each corner of the shape and label the new points with primes.\[A(x, y) \mapsto A'(\dots)\]
- Check the distances. Each new point should be the same distance from the center as the original point was.\[|OA| = |OA'|\]
Worked example
Triangle ABC has vertices A(3, 1), B(5, 4), C(2, 6). Rotate the triangle 90 degrees counterclockwise about the origin and find the image vertices.
Step 1: The center is the origin and the angle is 90 degrees counterclockwise.
Step 2: The rule is (x, y) becomes (-y, x).
Step 3: A(3, 1) becomes A'(-1, 3).
Step 4: B(5, 4) becomes B'(-4, 5).
Step 5: C(2, 6) becomes C'(-6, 2).
Check: The distance from the origin to A is the square root of (3^2 + 1^2) = the square root of 10. The distance from the origin to A' is the square root of ((-1)^2 + 3^2) = the square root of 10. They are the same.
The new vertices are A'(-1, 3), B'(-4, 5), and C'(-6, 2).
Tips and common mistakes
- Counterclockwise is the positive direction in math.
- 90 degrees counterclockwise is the same as 270 degrees clockwise. Be sure which direction you are turning.
- A 180 degree turn is the same in either direction. Use (x, y) becomes (-x, -y).
- A rotation keeps the same size, shape, and direction. It is not flipped like a reflection.
- If the center is not the origin, slide the picture so the center is at the origin, rotate, and then slide it back.
- The center of rotation does not move.
- Two rotations around the same center add up to one rotation. Add the angles.
- To check, make sure at least one point is the same distance from the center before and after.