Question

the trapezoid klmn is a dilation of the trapezoid klmn. 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 original and dilated points

First, find coordinates of a point from original trapezoid \( KLMN \) and its corresponding point in dilated trapezoid \( K'L'M'N' \). Let's take point \( K \) and \( K' \).

• Coordinates of \( K \): From the graph, \( K = (-6, -6) \)
• Coordinates of \( K' \): \( K' = (-2, -2) \)

Step2: Calculate scale factor

Scale factor \( k \) is the ratio of the coordinates of the dilated point to the original point (since dilation is about the origin, we can check the ratio of \( x \)-coordinates or \( y \)-coordinates).
For \( x \)-coordinate: \( \frac{-2}{-6} = \frac{1}{3} \)
For \( y \)-coordinate: \( \frac{-2}{-6} = \frac{1}{3} \)
We can verify with another point, say \( N \) and \( N' \).

• \( N = (-6, 3) \) (wait, earlier mistake: looking again, \( N \) is at \( (-6, 3) \)? Wait no, original trapezoid: \( K \) is at \( (-6, -6) \), \( N \) is at \( (-6, 3) \)? Wait no, the green trapezoid: \( K \) is at \( (-6, -6) \), \( N \) is at \( (-6, 3) \)? Wait no, looking at the grid: \( K \) is at \( (-6, -6) \), \( N \) is at \( (-6, 3) \)? Wait no, the green points: \( K \) is at \( (-6, -6) \), \( N \) is at \( (-6, 3) \)? Wait no, actually, let's re-express:

Wait, original trapezoid \( KLMN \):

• \( K \): \( (-6, -6) \)
• \( N \): \( (-6, 3) \) (vertical line from \( K \) to \( N \), length \( 3 - (-6) = 9 \))

Dilated trapezoid \( K'L'M'N' \):

• \( K' \): \( (-2, -2) \)
• \( N' \): \( (-2, 1) \) (vertical line from \( K' \) to \( N' \), length \( 1 - (-2) = 3 \))

Ratio of lengths: \( \frac{3}{9} = \frac{1}{3} \).
Alternatively, using coordinates: for \( N \): \( N = (-6, 3) \), \( N' = (-2, 1) \).
\( x \)-ratio: \( \frac{-2}{-6} = \frac{1}{3} \), \( y \)-ratio: \( \frac{1}{3} = \frac{1}{3} \).
Thus, scale factor is \( \frac{1}{3} \).

Answer:

\(\frac{1}{3}\)