Question

Suppose s is the line with equation yequals6 and t is the line with equation yequalsnegative 4. Write the following composition of reflections as one translation.
Upper R Subscript s Baseline circle Upper R Subscript t
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Part 1
Upper R Subscript s Baseline circle Upper R Subscript tequalsUpper T Subscript left angle nothing comma nothing right angle

Answer 1

To find the composition of reflections \( R_s \circ R_t \) as a single translation, we first identify the lines given:

- \( s: y = 6 \)
- \( t: y = -4 \)

1. **Determine the Distance Between the Lines**:
The distance between the lines \( y = 6 \) and \( y = -4 \) can be calculated as:
\[
\text{Distance} = 6 - (-4) = 6 + 4 = 10
\]

2. **Direction of the Translation**:
Since we are dealing with reflections across horizontal lines, the composed transformation will result in a vertical translation.

3. **Calculate the Translation Vector**:
The composition of two reflections over parallel lines results in a translation equal to twice the distance between the lines in the direction from the second line (the line with a lower y-value) to the first line (the line with a higher y-value).

- From line \( t \) to line \( s \), the translation distance is \( 10 \), and since we are translating from \( t \) to \( s \), the translation vector is:
\[
\vec{T} = (0, 10)
\]

4. **Write the Translation**:
Therefore, the composition of the reflections as a single translation \( T \) can be expressed as:
\[
R_s \circ R_t = T(0, 10)
\]

So, the final answer is:
\[
R_s \circ R_t = T(0, 10)
\]