Question

△cde was translated down and right to form triangle △cde. which statements are true? check all that apply.
□ de ≅ de
□ ec ≅ ec
□ ed ≅ dc
□ cd ≅ de
□ dc ≅ dc
□ △cde ≅ △cde

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. Rigid transformations preserve side - lengths and angles. So, if \(\triangle CDE\) is translated to \(\triangle C'D'E'\), then \(\triangle CDE\cong\triangle C'D'E'\) (by the definition of translation as a rigid transformation). Corresponding sides of congruent triangles are congruent.

Step2: Analyze each option

• For \(\overline{DE}\cong\overline{D'E'}\): Since \(\triangle CDE\cong\triangle C'D'E'\), \(DE\) and \(D'E'\) are corresponding sides. So, \(\overline{DE}\cong\overline{D'E'}\).
• For \(\overline{EC}\cong\overline{E'C'}\): \(EC\) and \(E'C'\) are corresponding sides of \(\triangle CDE\) and \(\triangle C'D'E'\). So, \(\overline{EC}\cong\overline{E'C'}\).
• For \(\overline{ED}\cong\overline{D'C'}\): \(ED\) in \(\triangle CDE\) and \(D'C'\) in \(\triangle C'D'E'\) are not corresponding sides.
• For \(\overline{C'D'}\cong\overline{D'E'}\): \(C'D'\) and \(D'E'\) are sides of \(\triangle C'D'E'\) and are not necessarily equal (they are two sides of a triangle, and there is no information to suggest they are equal).
• For \(\overline{D'C'}\cong\overline{DC}\): \(D'C'\) and \(DC\) are corresponding sides of \(\triangle C'D'E'\) and \(\triangle CDE\). So, \(\overline{D'C'}\cong\overline{DC}\).
• For \(\triangle CDE\cong\triangle C'D'E'\): As translation is a rigid transformation, \(\triangle CDE\cong\triangle C'D'E'\)

Answer:

\(\overline{DE}\cong\overline{D'E'}\), \(\overline{EC}\cong\overline{E'C'}\), \(\overline{D'C'}\cong\overline{DC}\), \(\triangle CDE\cong\triangle C'D'E'\)