Question

the given transformations map △ qrs onto its image △ q’r’s’. • a dilation by a scale factor of 1 centered at the origin • (x,y) → (x−2,y+2) complete the statement to compare the side lengths of △ qrs and △ q’r’s’. since dropdown preserve distance and dropdown preserve distance, the side lengths of △ qrs dropdown the corresponding side lengths of △ q’r’s’. the dropdown options shown include all dilations, all dilations do not, dilations with a scale factor of 1.

Explanation:

Step1: Analyze Dilation

A dilation with a scale factor of 1 means the image is congruent to the original figure (since scaling by 1 doesn't change the size). So dilations with a scale factor of 1 preserve distance.

Step2: Analyze Translation

The transformation \((x,y)\to(x - 2,y + 2)\) is a translation. Translations are rigid transformations, which preserve distance (side lengths, angles, etc.).

Step3: Compare Side Lengths

Since dilations with a scale factor of 1 preserve distance and translations preserve distance, the side lengths of \(\triangle QRS\) are equal to the corresponding side lengths of \(\triangle Q'R'S'\).

For the first blank: The correct option is "dilations with a scale factor of 1" (because only dilations with scale factor 1 preserve distance; other dilations (scale factor ≠1) change size). For the second blank, translations (the \((x,y)\to(x - 2,y + 2)\) transformation) preserve distance. And the relationship between side lengths is equal.

Answer:

First blank: dilations with a scale factor of 1; Second blank: translations (the \((x,y)\to(x - 2,y + 2)\) transformation); Relationship: are equal to