Question

triangle klm was dilated according to the rule do.0.75 (x,y). what is true about the image δklm? select three options. do.0.75 (x,y) = (0.75x, 0.75y) 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: Analyze the dilation rule

The dilation rule \(D_{O,0.75}(x,y)\) means that each coordinate of the original point \((x,y)\) is multiplied by \(0.75\). So \(D_{O,0.75}(x,y)=(0.75x,0.75y)\) is correct.

Step2: Check parallelism

When a figure is dilated, corresponding line segments are parallel. So \(\overline{LM}\) is parallel to \(\overline{L'M'}\) is correct.

Step3: Analyze the length of segments

Since the scale factor \(k = 0.75<1\), the image is smaller than the pre - image. So \(KM\) is longer than \(K'M'\) (not shorter).

Step4: Analyze the distance from vertices to the origin

Since the scale factor \(k = 0.75<1\), the vertices of the image are closer to the origin than those of the pre - image.

Step5: Analyze the distance from \(M\) and \(M'\) to the origin

The distance from a point \((x,y)\) to the origin is \(d=\sqrt{x^{2}+y^{2}}\). For point \(M(-2,2)\), \(d_M=\sqrt{(-2)^{2}+2^{2}}=\sqrt{4 + 4}=\sqrt{8}\). For \(M'\) with coordinates \((-2\times0.75,2\times0.75)=(-1.5,1.5)\), \(d_{M'}=\sqrt{(-1.5)^{2}+1.5^{2}}=\sqrt{2.25+2.25}=\sqrt{4.5}\), and \(d_{M'}=0.75d_M\) (not half).

Answer:

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