Question

the trapezoid ijkl is a dilation of the trapezoid ijkl. 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 coordinates of original and dilated points

Original trapezoid \( IJKL \): Let's take points \( I(-8, 0) \), \( J(8, 0) \), \( K(8, -8) \), \( L(0, -8) \).
Dilated trapezoid \( I'J'K'L' \): Points \( I'(-2, 0) \), \( J'(2, 0) \), \( K'(2, -2) \), \( L'(0, -2) \).

Step2: Calculate scale factor (ratio of corresponding side lengths)

Take the length of \( IJ \) (original) and \( I'J' \) (dilated).
Length of \( IJ \): \( |8 - (-8)| = 16 \)? Wait, no, wait. Wait, \( I(-8,0) \), \( J(8,0) \): distance is \( 8 - (-8) = 16 \)? Wait, no, wait the graph: Wait, \( I \) is at \( (-8,0) \), \( J \) at \( (8,0) \), but \( K \) is at \( (8, -8) \), \( L \) at \( (0, -8) \). Wait, maybe better to take vertical side \( JK \): \( J(8,0) \) to \( K(8, -8) \): length is \( |0 - (-8)| = 8 \).
Dilated \( J'K' \): \( J'(2,0) \) to \( K'(2, -2) \): length is \( |0 - (-2)| = 2 \).

Scale factor \( = \frac{\text{length of dilated side}}{\text{length of original side}} = \frac{2}{8} = \frac{1}{4} \). Wait, wait, no: Wait \( I'J' \): \( I'(-2,0) \), \( J'(2,0) \): length is \( 2 - (-2) = 4 \). Original \( IJ \): \( 8 - (-8) = 16 \)? No, wait the x-coordinates: \( I(-8,0) \), \( J(8,0) \): distance is \( 8 - (-8) = 16 \)? But \( I'(-2,0) \), \( J'(2,0) \): distance is \( 2 - (-2) = 4 \). So \( 4/16 = 1/4 \). Alternatively, take \( JK \): original \( JK \) is from \( (8,0) \) to \( (8, -8) \): length 8. Dilated \( J'K' \) from \( (2,0) \) to \( (2, -2) \): length 2. So \( 2/8 = 1/4 \). Yes, that's consistent.

Answer:

\(\frac{1}{4}\)