Question

nilar figures
triangle klm was dilated according to the rule $d_{o,0.75}(x, y)$.
what is true about the image $\triangle klm$? choose three correct answers.
$d_{o, 0.75}(x, y) = (0.75x, 0.75y)$
$\overline{km}$ is shorter than $\overline{km}$.
the vertices of the image are closer to the origin than those of the pre - image.
$\overline{lm}$ is parallel to $\overline{lm}$.

Explanation:

Step1: Analyze Dilation Rule

Dilation rule \( D_{O,0.75}(x,y)=(0.75x,0.75y) \) means each coordinate is scaled by 0.75 (a reduction, since \( 0.75 < 1 \)). So this statement is true.

Step2: Compare Segment Lengths

Since dilation with scale factor \( 0.75 \) (less than 1) reduces lengths, \( \overline{K'M'} = 0.75 \times \overline{KM} \), so \( \overline{KM} \) is longer than \( \overline{K'M'} \). The statement " \( \overline{KM} \) is shorter than \( \overline{K'M'} \)" is false.

Step3: Distance from Origin

Scaling coordinates by \( 0.75 \) (closer to 0) means image vertices are closer to the origin than pre - image. This statement is true.

Step4: Parallelism in Dilation

Dilation preserves the slope of segments, so \( \overline{LM} \) and \( \overline{L'M'} \) have the same slope, hence they are parallel. This statement is true.

Step5: Analyze Last Option (Incomplete, but from context)

The scale factor is \( 0.75 \) (not \( 0.5 \)), so the distance from \( M' \) to origin is \( 0.75 \) times that of \( M \), not half. So this statement (if about being half) is false.

Answer:

1. \( D_{O,0.75}(x, y)=(0.75x, 0.75y) \)
2. The vertices of the image are closer to the origin than those of the pre - image.
3. \( \overline{LM} \) is parallel to \( \overline{L'M'} \)