Question

consider the reflection of δabc across the line of reflection, (overleftrightarrow{pt}). which statements must be true? check all that apply. (square) (aa = cc) (square) (cq = qc) (square) (overleftrightarrow{pt} perp overline{aa}) (square) (overline{cc} perp overline{bb}) (square) (overline{aa} parallel overline{bb}) (square) (mangle trb = 90^circ)

Explanation:

To solve this, we use the properties of reflections:

• A reflection over a line \( \overleftrightarrow{PT} \) means the line \( \overleftrightarrow{PT} \) is the perpendicular bisector of the segment joining a point and its image (e.g., \( \overline{AA'} \), \( \overline{BB'} \), \( \overline{CC'} \)).
• Corresponding segments (like \( \overline{AA'} \), \( \overline{BB'} \), \( \overline{CC'} \)) are all perpendicular to the line of reflection \( \overleftrightarrow{PT} \), so they are parallel to each other.

Analyzing each statement:

1. \( A'A = C'C \): Not necessarily true. The lengths depend on the original positions, but reflection preserves distance from the line, not necessarily the length of the segment joining the point and its image.
2. \( C'Q = QC \): Since \( Q \) is on \( \overleftrightarrow{PT} \), and \( \overleftrightarrow{PT} \) is the perpendicular bisector of \( \overline{CC'} \), \( Q \) is equidistant from \( C \) and \( C' \), so \( C'Q = QC \). True.
3. \( \overleftrightarrow{PT} \perp \overline{AA'} \): By the property of reflection, the line of reflection is perpendicular to the segment joining a point and its image. So \( \overleftrightarrow{PT} \perp \overline{AA'} \). True.
4. \( \overline{C'C} \perp \overline{B'B} \): Since both \( \overline{C'C} \) and \( \overline{B'B} \) are perpendicular to \( \overleftrightarrow{PT} \), they are parallel, not perpendicular. False.
5. \( \overline{AA'} \parallel \overline{BB'} \): Both are perpendicular to \( \overleftrightarrow{PT} \), so they are parallel. True.
6. \( m\angle TRB = 90^\circ \): \( \overline{BB'} \) is perpendicular to \( \overleftrightarrow{PT} \) (by reflection property), so \( \angle TRB = 90^\circ \). True.

Answer:

B. \( C'Q = QC \)
C. \( \overleftrightarrow{PT} \perp \overline{AA'} \)
E. \( \overline{AA'} \parallel \overline{BB'} \)
F. \( m\angle TRB = 90^\circ \)