How to Find a Coordinate: Dilation

Coordinate dilation is a transformation that alters the size of a figure without changing its shape or orientation. The process scales up or down the distances between all points of a shape from a fixed point, known as the center of dilation.

A Step-by-step Guide to Finding a Coordinate: Dilation

Step 1: Identify the Center of Dilation

The first step in performing a coordinate dilation is to identify the center of dilation. This point acts as a fulcrum from which all other points within the shape will be dilated. In a coordinate plane, this can be any point, but it is typically chosen for convenience.

Step 2: Determine the Scale Factor

The next step involves determining the scale factor, denoted by the symbol ‘\(k\)’. The scale factor indicates how much the figure will be enlarged or reduced. If ‘\(k\)’ is greater than \(1\), the figure will be enlarged; if it’s between \(0\) and \(1\), the figure will be reduced; if ‘\(k\)’ is negative, the figure will be enlarged and reflected.

Step 3: Perform the Dilation

With the center of dilation and scale factor identified, you can now proceed with the dilation. This involves multiplying the coordinates of each point in the shape by the scale factor. It’s important to remember that the coordinates are measured from the center of dilation.

Step 4: Plot the Dilated Figure

After performing the dilation, the final step is to plot the new points on the coordinate plane. Connect the points to complete the dilated figure. Note that the shape and orientation remain unchanged, but the size of the figure has been altered according to the scale factor.

An Example of Coordinate Dilation

Consider a square with vertices at points \((2,2), (4,2), (4,4)\), and \((2,4)\). We choose the origin \((0,0)\) as our center of dilation and a scale factor of \(2\). We multiply the coordinates of each point by the scale factor to get our new points: \((4,4), (8,4), (8,8)\), and \((4,8)\). Plotting these on the coordinate plane, we see our original square has been dilated to twice its original size.

Conclusion

Coordinate dilation is a straightforward yet powerful concept in geometry, enabling us to change the size of shapes while maintaining their form and orientation. Through careful identification of the center of dilation, thoughtful selection of a scale factor, and diligent application of these elements, we can successfully dilate coordinates on a plane. It is a skill that not only benefits students in solving geometry problems but also professionals in fields such as computer graphics and architectural design, where the understanding of space, scale, and proportion is vital.