Question

determine if δxyz and δxyz are similar.
a. the triangles are similar because the dilation (x,y) → (x + 2,y + 2) maps δxyz to δxyz.
b. the triangles are similar because the dilation (x,y) → (2x,2y) maps δxyz to δxyz.
c. the triangles are not similar because the dilation (x,y) → (x + 2,y + 2) maps δxyz to δxyz.
d. the triangles are not similar because the dilation (x,y) → (2x,2y) maps δxyz to δxyz.

Explanation:

Step1: Recall Dilation Properties

Dilation is a transformation that scales a figure by a scale factor, preserving shape (similarity). Translations (like \((x,y)\to(x + 2,y + 2)\)) move figures, not scale. A dilation has the form \((x,y)\to(kx,ky)\) where \(k\) is the scale factor.

Step2: Analyze Option A and C

The transformation \((x,y)\to(x + 2,y + 2)\) is a translation, not dilation. So A and C are incorrect as they misidentify the transformation type.

Step3: Analyze Option B and D

The transformation \((x,y)\to(2x,2y)\) is a dilation with scale factor \(2\). Dilation preserves similarity, so the triangles should be similar. Thus, B is correct (triangles are similar due to dilation \((x,y)\to(2x,2y)\)) and D is incorrect.

Answer:

B. The triangles are similar because the dilation \((x,y)\to(2x,2y)\) maps \(\triangle XYZ\) to \(\triangle X'Y'Z'\).