Question

if \\( \triangle abc \\) is dilated from point a by a scale factor of \\( \frac { 1 } { 2 } \\), which of the following equations is true about \\( \overline { d ^ { \prime } e ^ { \prime } } \\)?

Explanation:

Step1: Recall the property of dilation

When a figure is dilated, the length of each segment in the image is equal to the scale factor multiplied by the length of the corresponding segment in the original figure.

Step2: Apply the dilation formula

Given the scale factor \(k = \frac{1}{2}\). For a segment \(\overline{DE}\) in the original figure and its corresponding segment \(\overline{D'E'}\) in the dilated figure, the formula is \(|\overline{D'E'}|=k\times|\overline{DE}|\). Substituting \(k = \frac{1}{2}\), we get \(|\overline{D'E'}|=\frac{1}{2}|\overline{DE}|\).

Answer:

\(\overline{D'E'}=\frac{1}{2}\overline{DE}\)