How to rotate a shape on a coordinate plane (Grade 8 steps)

📚 Understanding Rotations on a Coordinate Plane

Rotating a shape on a coordinate plane means turning it around a fixed point, called the center of rotation. We usually rotate shapes around the origin (0,0). The amount we turn it is measured in degrees. Common rotations are 90°, 180°, and 270°, either clockwise or counterclockwise. Let's break it down:

📐 Key Concepts

• 📍 Coordinate Plane: The grid made up of the x-axis (horizontal) and the y-axis (vertical). Points are located using ordered pairs (x, y).
• 🔄 Rotation: Turning a shape a certain number of degrees around a point. The size and shape stay the same; only the orientation changes.
• 🧭 Center of Rotation: The fixed point around which the shape rotates. Usually the origin (0,0).
• 🌡️ Degrees of Rotation: How much the shape is turned, measured in degrees (e.g., 90°, 180°, 270°).
• ➡️ Clockwise: Rotation in the same direction as the hands of a clock.
• ⬅️ Counterclockwise: Rotation in the opposite direction as the hands of a clock.

✏️ Steps for Rotating a Shape

Let's say we want to rotate a triangle with vertices A(1, 2), B(4, 2), and C(1, 5) by 90° counterclockwise around the origin.

1. 🗺️ Identify the Coordinates: Write down the coordinates of each vertex of the shape. In our example, A(1, 2), B(4, 2), and C(1, 5).
2. 📐 Apply the Rotation Rule: Use the appropriate rule for the rotation.
   • 90° Counterclockwise: (x, y) becomes (-y, x)
   • 180° Rotation: (x, y) becomes (-x, -y)
   • 270° Counterclockwise (same as 90° Clockwise): (x, y) becomes (y, -x)
3. ➕ Calculate New Coordinates: Apply the rule to each point.
   • For 90° counterclockwise:
   • A(1, 2) becomes A'(-2, 1)
   • B(4, 2) becomes B'(-2, 4)
   • C(1, 5) becomes C'(-5, 1)
4. 📈 Plot the New Points: Plot the new coordinates (A', B', C') on the coordinate plane.
5. 🔗 Connect the Points: Connect the plotted points to form the rotated shape.

✍️ Examples

Example 1: 180° Rotation

Rotate the point (3, -2) by 180° around the origin.

• Original point: (3, -2)
• Rule: (x, y) becomes (-x, -y)
• Rotated point: (-3, 2)

Example 2: 270° Counterclockwise Rotation

Rotate the point (-1, 4) by 270° counterclockwise around the origin.

• Original point: (-1, 4)
• Rule: (x, y) becomes (y, -x)
• Rotated point: (4, 1)

🧠 Practice Quiz

1. Rotate the point (2,3) 90 degrees clockwise.
2. Rotate the point (-1, -4) 180 degrees.
3. Rotate the point (5, -2) 90 degrees counter-clockwise.
4. What are the new coordinates of the point (0,4) when rotated 270 degrees counterclockwise?
5. Rotate a square ABCD with A(1,1) B(3,1) C(3,3) D(1,3) by 90 degrees clockwise around the origin. What are the new coordinates for A'?
6. After a rotation of 180 degrees, the image of point X is X'(-2, 5). What were the original coordinates of X?

Answers:

1. (3, -2)
2. (1, 4)
3. (2, 5)
4. (4, 0)
5. (1, -1)
6. (2, -5)