The Centre of Enlargement: A Simple Guide

The Centre of Enlargement is an important idea in geometry, especially when we are talking about enlarging or shrinking shapes.

So, what is it?

It is the fixed point in space around which a shape gets bigger or smaller. Here’s how it works:

1. Scale Factor:

   • The scale factor (we call it $k$) tells us how much the shape changes size.
   • If we have a point $P$ (the original point) and a point $P'$ (the new, transformed point), we can find the scale factor with this formula:
     $k = \frac{d(P', C)}{d(P, C)}$
   • In this formula, $C$ is the Centre of Enlargement, and $d(P, C)$ means the distance from $P$ to $C$.

2. Direction and Size:

   • The Centre of Enlargement also influences how the shape grows or shrinks.
   • If $k > 1$, it means the shape gets larger. If $0 < k < 1$, the shape becomes smaller.

3. Example:

   • Let’s say we have a triangle with points at $A(2, 3)$, $B(4, 5)$, and $C(6, 7)$.
   • If our Centre of Enlargement is $O(0, 0)$ and we use a scale factor of $2$, the new points will be:
     • $A'(4, 6)$
     • $B'(8, 10)$
     • $C'(12, 14)$
   • This shows how the triangle has grown bigger.

4. Real-life Use:

   • Knowing about enlargements is useful in many areas, like architecture, city planning, and digital design.
   • It helps make sure that shapes are scaled correctly, which is very important for creating things that fit well together.

Overall, the Centre of Enlargement helps us understand and work with changes in shape size in a clear and simple way!