Geometric Invariance Math Example 2

Solution

1. 1
   Step 1: Every point $(x,y)$ maps to $(kx, ky)$. The new centre is $(ka, kb)$ and the new radius is $|k|r$.
2. 2
   Step 2: The image is still a circle, so the property of being a circle (the shape) is invariant.
3. 3
   Step 3: The radius changes from $r$ to $|k|r \neq r$ (since $k\neq 1$). NOT invariant.
4. 4
   Step 4: The ratio $r/\sqrt{a^2+b^2}$ becomes $|k|r / (|k|\sqrt{a^2+b^2}) = r/\sqrt{a^2+b^2}$. This ratio IS invariant.
5. 5
   Step 5: Angles subtended at the origin by any arc or chord are preserved because dilation preserves angles.