Short Answer 
Dilation is a geometric transformation that changes the size of a figure while maintaining its shape, defined by a scale factor. In the given example, a scale factor of 1/3 and a center at (0, 0) will reposition the vertices of polygon MNPQ closer to the center by one-third of their distance, resulting in a smaller polygon M’N’P’Q’. The new coordinates can be calculated by multiplying the original coordinates by the scale factor.
Step 1: Understand Dilation
Dilation is a transformation that alters the size of a geometric figure while retaining its shape. This transformation is defined by a scale factor, denoted as ‘k’, which determines how much larger or smaller the new image will be compared to the original figure. For example, if the scale factor is greater than 1, the figure enlarges, while a factor between 0 and 1 reduces its size.
Step 2: Identify the Scale Factor and Center
In this case, we are given a scale factor of 1/3 and a center of dilation at coordinates (0, 0). This means that each point of the original polygon MNPQ will be repositioned closer to the center (0, 0) by one-third of its original distance. The scale factor affects every vertex of the polygon, creating a proportional image, or M’N’P’Q’.
Step 3: Calculate the New Coordinates
To find the new positions of the vertices after dilation, use the formula: new point = center + (original point – center) x scale factor. For example, if the original point is at coordinates (x, y), the new coordinates after applying the scale factor would be computed as follows:
- New x-coordinate = (1/3) * x
- New y-coordinate = (1/3) * y
By applying this formula to each vertex of polygon MNPQ, you’ll successfully transform it into the smaller polygon M’N’P’Q’.