Question

the triangle mno is a dilation of the triangle mno. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Identify coordinates of points

First, we find the coordinates of corresponding points in triangle \( MNO \) and \( M'N'O' \). Let's take point \( O \) and \( O' \), \( N \) and \( N' \), \( M \) and \( M' \).

• For \( O \): From the graph, \( O \) is at \( (-4, 4) \).
• For \( O' \): \( O' \) is at \( (-6, 6) \).

Step2: Calculate the scale factor

The scale factor \( k \) of a dilation is given by the ratio of the distance from the center of dilation (assuming the center is the origin or we can use the ratio of corresponding coordinates since dilation is about a common center, likely the origin here as the lines from \( M \) to \( M' \), \( N \) to \( N' \), \( O \) to \( O' \) seem to pass through the origin) of the image point to the pre - image point.

For the \( x \) - coordinate (or \( y \) - coordinate, since it's a dilation) of \( O \) and \( O' \):

The \( x \) - coordinate of \( O \) is \( - 4 \), the \( x \) - coordinate of \( O' \) is \( - 6 \). The ratio of \( x \) - coordinates (or \( y \) - coordinates) is \( \frac{-6}{-4}=\frac{3}{2} \). Let's verify with another point.

Take point \( N \): \( N \) is at \( (-4,-2) \), \( N' \) is at \( (-6,-3) \). The ratio of \( x \) - coordinates is \( \frac{-6}{-4}=\frac{3}{2} \), and the ratio of \( y \) - coordinates is \( \frac{-3}{-2}=\frac{3}{2} \).

Take point \( M \): \( M \) is at \( (2,-4) \), \( M' \) is at \( (3,-6) \). The ratio of \( x \) - coordinates is \( \frac{3}{2} \), and the ratio of \( y \) - coordinates is \( \frac{-6}{-4}=\frac{3}{2} \).

So the scale factor \( k=\frac{3}{2} \).

Answer:

\(\frac{3}{2}\)