Question

question 5 (multiple choice worth 1 point) (02.01p hc) triangle abc is translated by the rule (x + 1, y + 1) and then dilated by a scale factor of 2 centered at the origin. which statement describes the properties of triangles abc and a^b^c^ after the transformations? question 6 (multiple choice worth 1 point) (02.01 lc) ∠c and ∠c^ are congruent after both the translation and the dilation ∠c and ∠c^ are congruent after the dilation, but not after the translation ac and a^c^ are congruent after both the translation and the dilation ac and a^c^ are congruent after the dilation, but not after the translation

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. It preserves the shape and size of the figure. So, corresponding angles and corresponding sides of the original triangle \(ABC\) and the translated triangle \(A'B'C'\) are congruent. That is, if we only consider the translation \((x + 1,y+1)\), \(\angle C\cong\angle C'\) and \(\overline{AC}\cong\overline{A'C'}\).

Step2: Properties of dilation

Dilation is a non - rigid transformation. When we dilate a figure with a scale factor \(k = 2\) centered at the origin, the lengths of the sides of the figure change. If the original length of a side is \(l\), the length of the corresponding side after dilation is \(l'=kl\). So, \(\overline{A'C'}\) (after translation) has length \(l\) (same as \(\overline{AC}\)), and \(\overline{A''C''}\) (after dilation) has length \(l'' = 2l\). So, \(\overline{AC}\) and \(\overline{A''C''}\) are not congruent. But for angles, dilation preserves the measure of angles. Because the measure of an angle is determined by the amount of rotation between two rays forming the angle, and dilation only changes the length of the sides (rays) proportionally. So, \(\angle C\) and \(\angle C''\) are congruent after both the translation and the dilation.

Answer:

\(\angle C\) and \(\angle C''\) are congruent after both the translation and the dilation.