The Centre of Enlargement is an important idea in math. It helps us understand how shapes get bigger or smaller.

What is the Centre of Enlargement?

• Centre of Enlargement: This is a fixed point, often called O, that we use as a reference when we enlarge or reduce a shape.
• When we enlarge a shape, every point on that shape moves away from the centre O in a straight line based on a scale factor.

Why is the Centre of Enlargement Important?

1. Helps Us Transform Shapes Accurately:

   • The centre of enlargement tells us how a shape's size and position will change. Without a centre, it would be hard to make shapes bigger or smaller correctly.

2. Understanding Scale Factors:

   • A scale factor shows how much we are changing the size of the shape.
   • If the scale factor is more than 1, the shape gets bigger. If it's less than 1, the shape gets smaller. For example:
     • If the scale factor is 2, the shape doubles in size.
     • If it's 0.5, the shape is cut in half.

3. How to Calculate New Points:

   • If we have a point M with coordinates (x, y) that we want to enlarge from the centre O with coordinates (x_c, y_c) using a scale factor k, we can find the new coordinates M' like this:
     • M'(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))

4. Changing Shapes:

   • The centre of enlargement is really important when we change shapes, like enlarging a triangle. The corners, or vertices, of the triangle move out from the centre, keeping the shape balanced.

5. Everyday Uses:

   • We see the centre of enlargement in action in areas like architecture, graphic design, and making maps. It helps ensure everything stays in proportion and looks right.

Simple Examples

• Example 1: Imagine we have a triangle with points A(1, 2), B(3, 4), and C(5, 6). If the centre of enlargement is O(0, 0) and we want to use a scale factor of 2, the new points after enlarging would be:

  • A'(2, 4), B'(6, 8), C'(10, 12).

• Example 2: Now, let's reduce a square with points D(2, 2), E(2, 4), F(4, 4), and G(4, 2). If we use O(2, 2) as the centre and a scale factor of 0.5, the new points would be:

  • D'(2, 2), E'(2, 3), F'(3, 3), and G'(3, 2).

Knowing about the centre of enlargement is really important for Year 8 students. It helps them build a strong foundation in geometry and transformations, giving them skills they can use in many situations.