Question

the following is true for two triangles: def ~ abc. identify the true statements from the following set of options. select two correct answers.

Explanation:

Step1: Analyze similar triangles' side correspondence

For similar triangles \(DEF\sim ABC\), the order of the letters gives the correspondence of vertices \(D\) to \(A\), \(E\) to \(B\), \(F\) to \(C\). So the sides correspond as \(\overline{DE}\sim\overline{AB}\), \(\overline{EF}\sim\overline{BC}\), \(\overline{FD}\sim\overline{CA}\).

Step2: Check each option

• For \(\overline{CA}\sim\overline{EF}\): Incorrect as per correspondence.
• For \(\overline{FD}\sim\overline{AB}\): Incorrect as per correspondence.
• For \(\overline{BC}\sim\overline{EF}\): Incorrect (should be \(\overline{EF}\sim\overline{BC}\) but similarity of segments is commutative in a sense of proportion, but the order - if we consider the vertex - based correspondence, the correct correspondence for sides is based on vertex order. However, if we just consider the similarity of line - segments (as geometric objects), \(\overline{BC}\sim\overline{EF}\) (since similar triangles have proportional sides and \(\frac{BC}{EF}=\frac{AB}{DE}=\frac{AC}{DF}\)). But for vertex - based side - to - side similarity (where we map vertices), \(\overline{EF}\sim\overline{BC}\). But if we consider the general property of similar triangles (ratios of sides are equal), and the symbol \(\sim\) for segments (as in proportion), \(\overline{BC}\sim\overline{EF}\) (since \(\frac{BC}{EF}=k\), the scale factor).
• For \(\overline{BC}\sim\overline{FD}\): Incorrect as per correspondence.
• For \(\overline{FD}\sim\overline{CA}\): Correct as per vertex - based side correspondence (\(F\) corresponds to \(C\) and \(D\) corresponds to \(A\))

Answer:

\(\overline{BC}\sim\overline{EF}\), \(\overline{FD}\sim\overline{CA}\)