Consider the reflection of ΔABC across the line of reflection, Line P T.

Question

2 right triangles are shown. Line P T is the line of reflection. Line segment A prime A has a midpoint at point S. Line segment B prime B has a midpoint at point R. Line segment C prime C has a midpoint at point Q.
Which statements must be true? Check all that apply.

A'A = C'C
C'Q = QC
Line P T⊥ A'A
C'C ⊥ B'B
A'A || B'B
m∠TRB = 90°

Answer 1

When a triangle ΔABC is reflected across a line of reflection (Line PT), the following properties and statements about the reflections typically hold true:

1. A'A = C'C: This does not need to be true in general. The lengths of the segments from a point to its reflection can vary and are not necessarily equal.

2. C'Q = QC: This statement is true. Q is the midpoint of the segment C'C, and since Q is located on the line of reflection, C' will be the same distance from the line of reflection as C.

3. Line PT ⊥ A'A: This statement is true. By the property of reflection, a line of reflection is perpendicular to the segment joining each point and its reflected image.

4. C'C ⊥ B'B: This statement is not necessarily true. The orientation and position of triangles will determine the angles between segments after reflection.

5. A'A || B'B: This statement is true. Reflecting the triangles across the same line preserves the parallelism of corresponding segments between the original and reflected triangles.

6. m∠TRB = 90°: This statement is true. Since R is the midpoint of B'B and segment BR lies along the line of reflection, where angles are formed, the angle would indeed be 90°.

In summary, the true statements are:

• C'Q = QC
• Line PT ⊥ A'A
• A'A || B'B
• m∠TRB = 90°