Question

question 1
triangle abc* is formed using the translation (x + 0,y + 2) and the dilation by a scale factor of 2 from the origin. which equation explains the relationship between $overline{ac}$ and $overline{a^{*}c^{*}}$?
$\frac{overline{ac}}{overline{a^{*}c^{*}}}=2$
$\frac{overline{a^{*}c^{*}}}{overline{ac}}=\frac{1}{2}$
$overline{ac}=\frac{overline{a^{*}c^{*}}}{2}$
$overline{a^{*}c^{*}}=\frac{overline{ac}}{2}$

Explanation:

Step1: Recall dilation property

Dilation by a scale - factor \(k\) from the origin multiplies the length of all line - segments in a figure by \(k\). Here, the scale factor \(k = 2\).

Step2: Analyze the relationship between original and dilated line - segments

If a line - segment \(\overline{AC}\) is dilated by a scale factor of \(2\) from the origin to get \(\overline{A'C'}\), then the length of \(\overline{A'C'}\) is \(2\) times the length of \(\overline{AC}\), i.e., \(|\overline{A'C'}|=2|\overline{AC}|\), which can be rewritten as \(\frac{|\overline{A'C'}|}{|\overline{AC}|}=2\) or \(\overline{A'C'}=2\overline{AC}\) (in terms of length), and \(\frac{\overline{AC}}{\overline{A'C'}}=\frac{1}{2}\).

Answer:

\(\frac{\overline{AC}}{\overline{A'C'}}=\frac{1}{2}\) (equivalent to \(\overline{A'C'}=2\overline{AC}\)), so the correct option is \(\frac{\overline{A'C'}}{\overline{AC}} = 2\) (assuming the options are in terms of the ratio of the dilated segment to the original segment). If we consider the given options in the order presented, the correct one is the first option \(\frac{\overline{AC}}{\overline{A'C'}}=\frac{1}{2}\) (re - written from \(\frac{\overline{A'C'}}{\overline{AC}} = 2\)).