Three congruent triangles marked with congruent corresponding sides are arranged on two overlapping circles. The top circle is larger. All triangles overlap the top circle.

Question

Charlie is using rigid transformations to justify the SSS Congruence Theorem. He started with △ABC and △DEF that have three pairs of congruent sides. He translated and then rotated △ABC to create △A"B"C" such that A"B"¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ coincides with DE¯¯¯¯¯¯¯¯ . He then constructed the circles shown in the image and labeled Points 1 and 2. Why are Points 1 and 2 significant?

Option #1: They are the only points in the plane that are both a distance EF away from point E and a distance DF away from point D .

Option #2: They are the only points in the plane that are both a distance ED away from point E and a distance FE away from point F .

Option #3: They are the only points in the plane that are both a distance EF away from point E and a distance FE away from point F .

(1 point)
Option #
best describes the significance of Points 1 and 2.

Answer 1

To determine the significance of Points 1 and 2 regarding the construction performed by Charlie, let's analyze the context given the SSS (Side-Side-Side) Congruence Theorem.

According to the SSS Congruence Theorem, if two triangles have three pairs of congruent corresponding sides, then the triangles are congruent. Based on Charlie's description, he is working with triangles \( \triangle ABC \) and \( \triangle DEF \) where he has established that \( AB \cong DE \), \( BC \cong EF \), and \( CA \cong FD \).

When Charlie translates and rotates \( \triangle ABC \) to create \( \triangle A"B"C" \) such that side \( A"B" \) coincides with side \( DE \), he is leveraging the congruence of the sides to show that there are specific distances maintained which relate to the positions of points \( D \), \( E \), and the sides \( EF \) and \( DF \).

Next, let’s evaluate the options with this understanding:

1. Option #1: "They are the only points in the plane that are both a distance EF away from point E and a distance DF away from point D."

   • This option describes distances related to point \( E \) and segment \( DF \), which does not align with the congruence established and the points in context.

2. Option #2: "They are the only points in the plane that are both a distance ED away from point E and a distance FE away from point F."

   • This option also does not make sense in the context of the relationships in the problem as it incorrectly relays distances that are not congruent sides given that we are discussing distances concerning the sides \( EF \) and \( DF \).

3. Option #3: "They are the only points in the plane that are both a distance EF away from point E and a distance FE away from point F."

   • This option correctly states that Points 1 and 2 represent points that are at distances equal to \( EF \) from point \( E \) and \( FE \) from point \( F\). This is significant because it indicates potential locations for a point that maintains the congruent side lengths, reinforcing how \( E \) and \( F \) relate to constructing congruent triangles based on the sides identified.

Given this analysis, the most appropriate answer that describes the significance of Points 1 and 2 is:

Option #3. They are the only points in the plane that are both a distance EF away from point E and a distance FE away from point F.