Question

a reflection across \\(\overline{qr}\\) followed by a rotation \\(180^\circ\\) about point \\(q\\) maps equilateral triangle \\(qrs\\) onto triangle \\(qrs\\).
select all of the true statements.

• the position of point \\(q\\) is the same as the position of point \\(q\\).
• \\(\triangle qrs \cong \triangle qrs\\)
• \\(\overline{qr} \cong \overline{rs} \cong \overline{sq}\\)
• the position of point \\(r\\) is the same as the position of point \\(r\\).

Explanation:

1. Point Q's position: A reflection across $\overline{QR}$ leaves Q unchanged, and a 180° rotation about Q also leaves Q in place. So Q and Q' coincide.
2. Triangle congruence: Reflections and rotations are rigid transformations, which preserve size and shape, so $\triangle QRS \cong \triangle Q'R'S'$.
3. Side congruence: Since $\triangle QRS$ is equilateral, all sides are equal. Rigid transformations preserve side lengths, so $\overline{Q'R'} \cong \overline{R'S'} \cong \overline{S'Q'}$.
4. Point R's position: Reflecting R across $\overline{QR}$ leaves R unchanged, but a 180° rotation about Q moves R to a new position, so R and R' do not coincide.

Answer:

• The position of point Q is the same as the position of point Q'.
• $\triangle QRS \cong \triangle Q'R'S'$
• $\overline{Q'R'} \cong \overline{R'S'} \cong \overline{S'Q'}$