To understand rotation in a coordinate plane, we need to break down a few important ideas.

We are going to look at how shapes can turn around a point on a coordinate system. This is important for Year 10 students who are learning for their GCSE exams.

What is Rotation?

Rotation is when a shape moves in a circle around a fixed point called the center of rotation.

The angle of rotation tells us how much the shape is turned. We usually measure this in degrees (like 90°) or radians.

Here are the key parts to think about when we talk about rotation:

• Center of Rotation: This is the point that stays still while the shape turns. It can be described using two numbers $(x_c, y_c)$, showing where it sits on the coordinate grid.

• Angle of Rotation: This shows how far we need to turn the shape. A positive angle means we turn the shape to the left (counterclockwise), while a negative angle means we turn it to the right (clockwise).

• Shape and Its Coordinates: Each corner or point of the shape can be found using its coordinates $(x, y)$.

Now that we know about these parts, we can use a formula to see how a shape changes when it rotates.

How to Rotate a Point

If we want to rotate a point $(x, y)$ around a center point $(x_c, y_c)$ by an angle $\theta$, we follow these steps:

1. Move the Point: First, we change the position of the point so that the center of rotation is at the origin (the point (0,0)). We do this by subtracting the center's coordinates from the point's coordinates: $(x', y') = (x - x_c, y - y_c)$

2. Rotate the Point: Next, we rotate the point using the angle $\theta$. The formulas to find the new position after rotation are: $x'' = x' \cos(\theta) - y' \sin(\theta)$ $y'' = x' \sin(\theta) + y' \cos(\theta)$

3. Move Back: Finally, we return the point to its original position: $(x_{new}, y_{new}) = (x'' + x_c, y'' + y_c)$

By following these steps, we can get a clear picture of how a shape rotates.

Example of Rotation

Let’s see an example to make this clearer.

Imagine we have a triangle with corners or points at $A(2, 3)$, $B(4, 5)$, and $C(3, 1)$. We want to rotate this triangle 90 degrees to the left around the point $P(2, 2)$.

Step 1: Move the Points

• For point A: $(x', y') = (2 - 2, 3 - 2) = (0, 1)$

• For point B: $(x', y') = (4 - 2, 5 - 2) = (2, 3)$

• For point C: $(x', y') = (3 - 2, 1 - 2) = (1, -1)$

Step 2: Rotate (90 degrees) For a 90-degree turn, we use the formulas:

• For Point A: $x'' = 0 \cdot 0 - 1 \cdot 1 = -1$ $y'' = 0 \cdot 1 + 1 \cdot 0 = 0$

• For Point B: $x'' = 2 \cdot 0 - 3 \cdot 1 = -3$ $y'' = 2 \cdot 1 + 3 \cdot 0 = 2$

• For Point C: $x'' = 1 \cdot 0 - (-1) \cdot 1 = 1$ $y'' = 1 \cdot 1 + (-1) \cdot 0 = 1$

Step 3: Move Back Now we put the points back in their original position:

• Point A: $(x_{new}, y_{new}) = (-1 + 2, 0 + 2) = (1, 2)$
• Point B: $(x_{new}, y_{new}) = (-3 + 2, 2 + 2) = (-1, 4)$
• Point C: $(x_{new}, y_{new}) = (1 + 2, 1 + 2) = (3, 3)$

So, after the rotation, the new points of the triangle are $A'(1, 2)$, $B'(-1, 4)$, and $C'(3, 3)$.

Visual Representation

Once we have the new points, students can draw both the original and the rotated shapes on a graph. This helps visualize how the rotation works and shows how the shape changed its position but kept the same size.

Using graphing tools or apps can make this even more exciting. They allow you to see the rotation happening in real-time, making it easier to understand.

Applications of Rotation

Knowing how to rotate shapes is useful in many real-life situations, including:

• Computer Graphics: Turning and designing characters in video games and animations.

• Engineering: Rotating parts in a design to fit them correctly.

• Physics: Looking at how things move, like in sports (think about a spinning basketball).

Conclusion

Understanding rotation in a coordinate plane helps us in school and in many jobs. By practicing how to rotate shapes and seeing their properties, students develop important problem-solving skills that will aid them in math and many other areas.

To sum it up, to visualize rotation:

• Know the center of rotation and the angle.
• Move the point, rotate it, and then move it back.
• Practice with different shapes and angles to learn more.
• Use graphing tools for a better understanding of how rotation changes a shape on the grid.

This knowledge is a stepping stone to more advanced concepts in geometry that students will encounter in the future.