Transformations in coordinate geometry are a fun way to see how shapes change when we do certain things to them. If you're in Year 10 and learning about transformations, it's important to know how to use coordinate geometry. Let's look at how we can understand these transformations step by step!

Types of Transformations

In coordinate geometry, there are four main types of transformations:

1. Translation: This means sliding a shape in any direction without changing how big or what direction it faces. For example, if we move the point $A(2, 3)$ by a vector $(3, -1)$, the new spot for point $A'$ will be $A'(5, 2)$.

2. Reflection: This transformation flips a shape over a line, like the x-axis or y-axis. If we reflect the point $B(4, 2)$ over the y-axis, we get the new point $B'(-4, 2)$.

3. Rotation: You can also rotate shapes around a point, usually the origin, by a certain angle. For instance, rotating point $C(1, 0)$ 90 degrees counterclockwise around the origin gives us $C'(0, 1)$.

4. Enlargement: This transformation changes the size of the shape but keeps its proportions the same. If we enlarge from the origin with a scale factor of 2, point $D(2, 3)$ becomes $D'(4, 6)$.

How to Apply Transformations with Coordinate Geometry

To use these transformations correctly, we need to follow some rules:

For Translation

If we want to move a point $(x, y)$ by a vector $(a, b)$, we find the new coordinates like this:

$(x', y') = (x + a, y + b)$

Example: If we translate the point $(2, 3)$ by $(4, 5)$, we get:

$(2', 3') = (2 + 4, 3 + 5) = (6, 8)$

For Reflection

When reflecting across the x-axis or y-axis, it's simple:

• Across the x-axis: $(x, y) \to (x, -y)$
• Across the y-axis: $(x, y) \to (-x, y)$

For Rotation

To rotate a point $(x, y)$ by an angle $\theta$, we can use these formulas (make sure to use radians for angles):

$(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$

Example: Rotating $(1, 0)$ by 90 degrees (or $\frac{\pi}{2}$ radians) gives:

$(x', y') = (1 \cdot 0 - 0 \cdot 1, 1 \cdot 1 + 0 \cdot 0) = (0, 1)$

For Enlargement

For enlarging from the origin with a scale factor $k$, the transformation can be shown as:

$(x', y') = (kx, ky)$

Example: An enlargement of point $(2, 3)$ with a scale factor of 3 gives:

$(x', y') = (3 \cdot 2, 3 \cdot 3) = (6, 9)$

Problem-Solving with Geometric Reasoning

These formulas help us solve all sorts of geometric problems. Whether you're trying to find new positions for shapes after different transformations or figuring out coordinates from given points, coordinate geometry is a great tool!

Here’s a problem to try:

Problem: Imagine you have a triangle with points at $A(1, 2)$, $B(3, 4)$, and $C(5, 2)$. First, reflect this triangle over the y-axis and then translate it using the vector $(-2, 1)$.

1. Reflection:

   • $A'(-1, 2)$, $B'(-3, 4)$, $C'(-5, 2)$

2. Translation:

   • $A''(-1 - 2, 2 + 1) = (-3, 3)$
   • $B''(-3 - 2, 4 + 1) = (-5, 5)$
   • $C''(-5 - 2, 2 + 1) = (-7, 3)$

So, the final points of the transformed triangle are $A''(-3, 3)$, $B''(-5, 5)$, and $C''(-7, 3)$.

In summary, coordinate geometry helps us explore transformations easily and sets us up for more complex problems in the future. Have fun transforming shapes!