Question

How far apart are parallel lines m and n such that T<0, –12>(ΔXYZ) = (Rn ∘ Rm)(ΔXYZ)?

Parallel lines m and n are
units apart.

Answer 1

Given the expression \( T<0, -12>(\Delta XYZ) = (R_n \circ R_m)(\Delta XYZ) \), where \( R_n \) and \( R_m \) are reflections across the lines \( n \) and \( m \) respectively, we are asked to find the distance between the parallel lines \( m \) and \( n \).

To solve this, we can consider the transformation described on a triangle \(\Delta XYZ\) through the composition of the two reflections.

1. **Properties of Reflection**: The composition of two reflections across parallel lines results in a translation. Specifically, if two lines \( m \) and \( n \) are parallel and the distance between them is \( d \), the composition \( R_n \circ R_m \) will translate points by a distance of \( 2d \) in the direction perpendicular to the lines.

2. **Transformation Given**: The notation \( T<0, -12> \) indicates a transformation that translates points vertically downward by 12 units.

3. **Equating Translations**: From the properties of reflection, we know:
\[
(R_n \circ R_m)(\Delta XYZ) = \text{Translation by } 2d.
\]
Since \( T<0, -12> \) corresponds to a vertical translation downward by 12 units, we set the distances equal:
\[
2d = 12.
\]

4. **Solving for \( d \)**:
\[
d = \frac{12}{2} = 6.
\]

Thus, the distance between the parallel lines \( m \) and \( n \) is \( \boxed{6} \) units.