Question

1. (03.01 mc)

triangle xyz is translated by the rule (x + 1, y − 1) and then dilated by a scale factor of 4 centered at the origin. which statement describes the properties of triangles xyz and xyz after the transformations? (

• y and ∠y are congruent after the translation, but not after the dilation.
• ∠y and ∠y are congruent after the dilation, but not after the translation.
• \\(overline{yz}\\) and \\(overline{yz}\\) are proportional after the translation, but not after the dilation.
• \\(overline{yz}\\) and \\(overline{yz}\\) are proportional after the dilation and congruent after the translation.

1. (03.01 hc)

image of a grid with quadrilaterals efgh and efgh
quadrilateral efgh was dilated by a scale factor of 2 from the center (1, 0) to create efgh.

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. Rigid transformations preserve the shape and size of the figure. So, after translation, corresponding angles and sides of the two triangles (the original and the translated one) are congruent. That is, if we consider \(\angle Y\) and \(\angle Y'\) (after translation), \(\angle Y\cong\angle Y'\), and for sides like \(\overline{YZ}\) and \(\overline{Y'Z'}\) (after translation), \(\overline{YZ}\cong\overline{Y'Z'}\).

Step2: Properties of dilation

Dilation is a non - rigid transformation. A dilation with a scale factor \(k
eq1\) changes the size of the figure but not the shape. The ratio of the lengths of corresponding sides of the original figure and the dilated figure is equal to the scale factor. If the scale factor \(k = 4\), for any two corresponding sides \(a\) (of the original triangle) and \(a''\) (of the dilated triangle), \(\frac{a''}{a}=4\). Also, the measures of corresponding angles of the original figure and the dilated figure are equal. Because the shape is preserved (the angles of a triangle are determined by the relative orientation of its sides, and dilation does not change the relative orientation of the sides). So, \(\angle Y\cong\angle Y''\) (after translation and dilation), and for sides \(\overline{YZ}\) and \(\overline{Y''Z''}\), \(\overline{Y''Z''}=4\overline{YZ}\) (proportional)

Answer:

\(\overline{YZ}\) and \(\overline{Y''Z''}\) are proportional after the dilation and congruent after the translation.