Alright, let's break down the statements given in the proof and identify the appropriate reasoning for statement 7.
1. Defining the vertices of [tex]\(\triangle ABC\)[/tex] with unique points [tex]\(A(x_1, y_1), B(x_2, y_2),\)[/tex] and [tex]\(C(x_3, y_3)\)[/tex].
- Reason: Given.
2. Using rigid transformations to transform [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A'B'C'\)[/tex] so that [tex]\(A'\)[/tex] is at the origin and [tex]\(A'C'\)[/tex] lies on the x-axis in the positive direction.
- Reason: In the coordinate plane, any point can be moved to any other point using rigid transformations and any line can be moved to any other line using rigid transformations.
3. Any property that is true for [tex]\(\triangle A'B'C'\)[/tex] will also be true for [tex]\(\triangle ABC\)[/tex].
- Reason: Definition of congruence.
4. Let [tex]\(r, s,\)[/tex] and [tex]\(t\)[/tex] be real numbers such that the vertices of [tex]\(\triangle A'B'C'\)[/tex] are [tex]\(A'(0,0), B'(2r, 2s),\)[/tex] and [tex]\(C'(2t, 0)\)[/tex].
- Reason: Defining constants.
5. Let [tex]\(D', E',\)[/tex] and [tex]\(F'\)[/tex] be the midpoints of [tex]\(\overline{A'B'}, \overline{B'C'},\)[/tex] and [tex]\(\overline{A'C'}\)[/tex] respectively.
- Reason: Defining points.
Examining the proof structure, after defining the points and transformations, the key point would be to dive into the properties of these midpoints. For example, analyzing the slopes to establish relationships for the midpoints.
6. Compute the slopes of sides and segments to use in further geometric proofs.
- Reason: Definition of slope.
Now, coming to statement 7, which usually follows from the establishment of slopes:
7. Deriving or proving that certain lines are parallel or have equal slopes.
- Reason: Parallel lines have equal slopes.
In statement 7, if we need to find the reason for a subsequent geometry property related to slopes, it logically follows that we are using the relationship that parallel lines have equal slopes.
Thus, the most appropriate reasoning for statement 7, based on given information, would be:
C. Parallel lines have equal slopes.
