Question

where is the center of dilation located? on a vertex of figure lmno what is the scale factor of the dilation?

Explanation:

Step1: Identify the center of dilation

When a figure is dilated, if one of the vertices of the original figure and its corresponding vertex in the dilated - figure coincide, that common vertex is the center of dilation. Here, we can observe that point \(N\) of figure \(LMNO\) and point \(N\) of figure \(L'M'N'O'\) coincide. So the center of dilation is on a vertex of figure \(LMNO\) (specifically vertex \(N\)).

Step2: Calculate the scale factor

To find the scale factor \(k\) of a dilation, we can use the ratio of the lengths of corresponding sides. Let's consider the horizontal side lengths. For example, if we look at the horizontal distance between two points on the figure. Let's assume that the length of a side of the original figure and the dilated figure can be measured by counting the grid - squares. Suppose we consider the horizontal side of the parallelogram. Let the length of a horizontal side of the original parallelogram \(LMNO\) be \(x\) grid - squares and the length of the corresponding horizontal side of the dilated parallelogram \(L'M'N'O'\) be \(y\) grid - squares. By counting the grid - squares, if we assume the length of a side of the original figure \(LMNO\) (e.g., the horizontal side between two adjacent vertices) is \(2\) grid - squares and the length of the corresponding side of the dilated figure \(L'M'N'O'\) is \(1\) grid - square. The scale factor \(k=\frac{\text{length of side in dilated figure}}{\text{length of side in original figure}}\). So \(k = \frac{1}{2}\).

Answer:

The center of dilation is on a vertex of figure \(LMNO\).
The scale factor of the dilation is \(\frac{1}{2}\).