Question

triangle klm was dilated according to the rule do, 0.75 (x,y)=(0.75x, 0.75y). what is true about the image δklm? select three options.
lm is parallel to lm.
km is shorter than km.
the vertices of the image are closer to the origin than those of the pre - image.
the distance from m to the origin is exactly half the distance from m to the origin.

Explanation:

Step1: Recall dilation properties

A dilation with a scale - factor \(k = 0.75\) (where \(0

Step2: Analyze side - length relationships

For any line segment in the pre - image and its corresponding line segment in the image after a dilation centered at the origin with scale - factor \(k\), the length of the line segment in the image is \(k\) times the length of the line segment in the pre - image. Since \(k = 0.75\), the lengths of the sides of the image are shorter than the lengths of the sides of the pre - image. So, \(\overline{KM}\) is shorter than \(\overline{KM'}\) is false. Also, corresponding line segments in a dilation centered at the origin are parallel. So, \(\overline{LM}\) is parallel to \(\overline{LM'}\).

Step3: Analyze distance from origin

The distance from a point \((x,y)\) to the origin is \(d=\sqrt{x^{2}+y^{2}}\), and the distance from the dilated point \((kx,ky)\) to the origin is \(d'=\sqrt{(kx)^{2}+(ky)^{2}}=|k|\sqrt{x^{2}+y^{2}}\). Since \(k = 0.75\), the vertices of the image are closer to the origin than those of the pre - image. The distance from \(M'\) to the origin is \(0.75\) times the distance from \(M\) to the origin, not exactly half.

Step4: Analyze the dilation rule

The dilation rule \((x,y)\to(0.75x,0.75y)\) is correct as it represents a dilation centered at the origin with a scale - factor of \(0.75\).

Answer:

The dilation rule \((x,y)=(0.75x,0.75y)\) is true.
\(\overline{LM}\) is parallel to \(\overline{LM'}\) is true.
\(\overline{KM}\) is shorter than \(\overline{KM'}\) is false.
The vertices of the image are closer to the origin than those of the pre - image is true.
The distance from \(M'\) to the origin is exactly half the distance from \(M\) to the origin is false.