Question

the rectangle lmno is a dilation of the rectangle lmno. 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: Find length of LO and L'O'

First, find the length of \( LO \) in rectangle \( LMNO \). The coordinates of \( L \) are \((-10, 0)\) and \( O \) are \((10, 0)\), so the length \( LO = 10 - (-10)=20\). For \( L'O' \), coordinates of \( L' \) are \((-4, 0)\) and \( O' \) are \((4, 0)\), so \( L'O' = 4 - (-4) = 8\).

Step2: Calculate scale factor

The scale factor of dilation is the ratio of the length of the image (\( L'O' \)) to the length of the original (\( LO \)). So scale factor \(=\frac{L'O'}{LO}=\frac{8}{20}=\frac{2}{5}\). Alternatively, we can check the vertical side. Length of \( LM \): \( L(-10,0) \) to \( M(-10, -5) \) (wait, looking at the graph, \( M \) is at \((-10, -5)\)? Wait, no, the y - axis: \( M \) is at \((-10, -5)\)? Wait, no, the grid: from \( L(-10,0) \) down to \( M(-10, -5) \)? Wait, no, the original rectangle \( LMNO \): \( L(-10,0) \), \( M(-10, -5) \)? Wait, no, looking at the graph, the vertical side: \( L \) is at \((-10,0)\), \( M \) is at \((-10, -5)\)? Wait, no, the \( y \) - coordinate of \( M \) is - 5? Wait, no, the grid lines: each square is 1 unit. So \( L(-10,0) \), \( M(-10, -5) \)? Wait, no, the original rectangle: \( L(-10,0) \), \( M(-10, -5) \), \( N(10, -5) \), \( O(10,0) \). So the length of \( LM \) (vertical side) is \( 0 - (-5)=5 \)? Wait, no, \( L \) is at \( y = 0 \), \( M \) is at \( y=-5 \), so length \( LM = 5 \). For the image \( L'M'N'O' \), \( L'(-4,0) \), \( M'(-4, -2) \), so length \( L'M' = 0 - (-2)=2 \). Then scale factor \(=\frac{L'M'}{LM}=\frac{2}{5}\), which matches the horizontal side ratio. So the scale factor is \(\frac{2}{5}\).

Answer:

\(\frac{2}{5}\)