Step 6 of the question requires us to find the coordinates of the midpoints [tex]\( P^{\prime}, Q^{\prime}, \)[/tex] and [tex]\( R^{\prime} \)[/tex] of the line segments [tex]\( \overline{M^{\prime}N^{\prime}}, \overline{N^{\prime}O^{\prime}}, \)[/tex] and [tex]\( \overline{M^{\prime}O^{\prime}} \)[/tex], respectively. To do this, we use the definition of a midpoint, which states that the coordinates of the midpoint of a line segment whose endpoints are [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Let's apply this formula to find [tex]\( P^{\prime}, Q^{\prime}, \)[/tex] and [tex]\( R^{\prime} \)[/tex]:
1. [tex]\( P^{\prime} \)[/tex] is the midpoint of [tex]\( \overline{M^{\prime}N^{\prime}} \)[/tex] with endpoints [tex]\( M^{\prime}(0, 0) \)[/tex] and [tex]\( N^{\prime}(2r, 2s) \)[/tex]:
[tex]\[
P^{\prime} = \left( \frac{0 + 2r}{2}, \frac{0 + 2s}{2} \right) = \left( r, s \right)
\][/tex]
2. [tex]\( Q^{\prime} \)[/tex] is the midpoint of [tex]\( \overline{N^{\prime}O^{\prime}} \)[/tex] with endpoints [tex]\( N^{\prime}(2r, 2s) \)[/tex] and [tex]\( O^{\prime}(2t, 0) \)[/tex]:
[tex]\[
Q^{\prime} = \left( \frac{2r + 2t}{2}, \frac{2s + 0}{2} \right) = \left( r + t, s \right)
\][/tex]
3. [tex]\( R^{\prime} \)[/tex] is the midpoint of [tex]\( \overline{M^{\prime}O^{\prime}} \)[/tex] with endpoints [tex]\( M^{\prime}(0, 0) \)[/tex] and [tex]\( O^{\prime}(2t, 0) \)[/tex]:
[tex]\[
R^{\prime} = \left( \frac{0 + 2t}{2}, \frac{0 + 0}{2} \right) = \left( t, 0 \right)
\][/tex]
Therefore, the reason that completes the proof for step 6 is the definition of midpoint.
So, the correct answer is:
C. Definition of midpoint
