Question

1. triangle abc was translated to create triangle edk using the transformation, (x,y)→(x−9,y−2). which of the following congruence statements does not accurately describe the two triangles? ke≅ca ∠kde≅∠cba ba≅de ∠dke≅∠abc

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures. So, if \(\triangle ABC\) is translated to \(\triangle EDK\), then corresponding sides and corresponding angles are congruent.
Let's find the correspondence of vertices.
If \((x,y)\to(x - 9,y - 2)\), assume \(A\to K\), \(B\to D\), \(C\to E\) (by analyzing the direction and magnitude of the translation vector \((-9,-2)\) from the graph).

Step2: Check each congruence statement

• For \(\overline{KE}\cong\overline{CA}\): Since \(C\to E\) and \(A\to K\) (by translation correspondence), \(KE\) and \(CA\) are corresponding sides. So, \(\overline{KE}\cong\overline{CA}\) (because translation preserves side - length).
• For \(\angle KDE\cong\angle CBA\): Since \(B\to D\) and \(C\to E\) and \(A\to K\), \(\angle KDE\) corresponds to \(\angle CBA\) (translation preserves angle - measure).
• For \(\overline{BA}\cong\overline{DE}\): Since \(B\to D\) and \(A\to K\) (not \(E\)), \(BA\) corresponds to \(DK\) (not \(DE\)). The correct correspondence for \(DE\) is \(BC\) (because \(B\to D\) and \(C\to E\)).
• For \(\angle DKE\cong\angle ABC\): Since \(A\to K\), \(B\to D\), \(C\to E\), \(\angle DKE\) corresponds to \(\angle BAC\) (incorrect). Wait, no, if we assume the correct vertex correspondence: \(\triangle ABC\cong\triangle KDE\) (by translation). Then \(\angle KDE\) corresponds to \(\angle ABC\), \(\overline{BA}\) corresponds to \(\overline{DK}\), \(\overline{BC}\) corresponds to \(\overline{DE}\), \(\overline{CA}\) corresponds to \(\overline{EK}\).

Answer:

\(\angle DKE\cong\angle ABC\)