Question

is quadrilateral jklm the result of a dilation of quadrilateral abcd by a scale factor of 2? why or why not?
no, because sides jm and kl have different slopes from sides ad and bc.
yes, because both figures are parallelograms, so corresponding sides are parallel.
no, because sides jk and ml are not twice as long as sides ab and dc.
yes, because sides jk and ml are twice as long as sides ab and dc.

Explanation:

Step1: Recall Dilation Properties

Dilation scales lengths by a factor, preserves parallelism, and corresponding sides should be scaled by the factor. For scale factor 2, corresponding sides of J K L M should be twice those of A B C D.

Step2: Analyze Side Lengths

• Side \( AB \): From \( (2, -2) \) to \( (4, -2) \), length \( 4 - 2 = 2 \).
• Side \( JK \): From \( (5, -4) \) to \( (9, -4) \), length \( 9 - 5 = 4 \). \( 4 = 2\times2 \), so \( JK = 2\times AB \).
• Side \( DC \): From \( (1, -4) \) to \( (3, -4) \), length \( 3 - 1 = 2 \).
• Side \( ML \): From \( (2, -9) \) to \( (6, -9) \), length \( 6 - 2 = 4 \). \( 4 = 2\times2 \), so \( ML = 2\times DC \).
• Check parallelism: \( AB \parallel DC \parallel JK \parallel ML \) (horizontal lines, slope 0), \( AD \parallel BC \parallel JM \parallel KL \) (same slope, as they are parallel sides of parallelograms). So dilation (scale factor 2) is valid as corresponding sides are scaled by 2 and parallelism is preserved. The option "Yes, because sides JK and ML are twice as long as sides AB and DC" is correct.

Answer:

Yes, because sides JK and ML are twice as long as sides AB and DC.