- A circle is defined by \((x-a)^2 + (y-b)^2 = r^2\).
- Applying a dilation of \(k = \frac{1}{r}\) centered at \((a, b)\) transforms this into \((x'-a)^2 + (y'-b)^2 = 1\).
- This result shows that any circle can be scaled to have a radius of 1.
- Conclude that all circles are similar to a circle of radius 1: \(\boxed{\text{All circles are similar to a circle with a radius of 1 unit.}}\)
Explanation
- Define the transformation and coordinates.
Consider a circle with the equation \((x-a)^2 + (y-b)^2 = r^2\). The center of this circle is \((a, b)\) and its radius is \(r\). A dilation with a scale factor \(k = \frac{1}{r}\) and center of dilation at \((a, b)\) maps any point \((x, y)\) on the circle to a new point \((x', y')\) according to the following transformation rules:
\[x' - a = \frac{1}{r}(x - a) \implies x - a = r(x' - a)\]
\[y' - b = \frac{1}{r}(y - b) \implies y - b = r(y' - b)\]
- Apply the dilation to the circle's equation.
Substitute these relationships back into the original circle's equation \((x - a)^2 + (y - b)^2 = r^2\):
\[(r(x' - a))^2 + (r(y' - b))^2 = r^2\]
Expanding the squared terms:
\[r^2(x' - a)^2 + r^2(y' - b)^2 = r^2\]
Divide both sides of the equation by \(r^2\):
\[(x' - a)^2 + (y' - b)^2 = 1\]
- Interpret the geometric result.
The resulting equation \((x' - a)^2 + (y' - b)^2 = 1\) represents a circle with the same center \((a, b)\) but with a radius of \(1\) unit. Because a dilation is a type of similarity transformation, this demonstrates that any circle of radius \(r\) can be mapped to a circle of radius \(1\) through a similarity transformation. Therefore, all circles are similar to a circle with a radius of \(1\) unit.
Examples
Think of mapping geographical features on a map. Regardless of the actual size of a circular lake on the ground (its radius \(r\)), when you draw it on a paper map as a unit circle (radius \(1\)), you are using the concept of similarity. This mathematical transformation confirms that all such circular shapes in geometry maintain their proportional properties regardless of their scale, just as all circles share the same fundamental shape characteristics as a unit circle.