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Sagot :
Certainly! Let's carefully examine the details and logic behind the given proof and reach the conclusion step-by-step to find the reason for statement 7.
Here's an overview of the proof provided:
1. Define the vertices of [tex]\(\triangle ABC\)[/tex] at specific points [tex]\(A (x_1, y_1)\)[/tex], [tex]\(B (x_2, y_2)\)[/tex], [tex]\(C(x_3, y_3)\)[/tex].
- Reason: Given.
2. Use rigid transformations to transform [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A'B'C'\)[/tex] such that [tex]\(A'\)[/tex] is at the origin, and [tex]\(\overline{A'C'}\)[/tex] lies along the x-axis.
- Reason: Using rigid transformations, any point can move to any other point, and any line can move to any other line.
3. Any property true for [tex]\(\triangle A'B'C'\)[/tex] is also true for [tex]\(\triangle ABC\)[/tex].
- Reason: Definition of congruence.
4. Use constants [tex]\(r, s, t\)[/tex] to redefine the vertices [tex]\(A'(0,0)\)[/tex], [tex]\(B'(2r, 2s)\)[/tex], [tex]\(C'(2t, 0)\)[/tex].
- Reason: Defining constants.
5. Define midpoints of [tex]\(\overline{A'B'}\)[/tex], [tex]\(\overline{B'C'}\)[/tex], and [tex]\(\overline{A'C'}\)[/tex] as [tex]\(D', E', F'\)[/tex].
- Reason: Defining points.
6. Specify the coordinates for [tex]\(D'\)[/tex] and [tex]\(F'\)[/tex] as [tex]\(D'(r, s)\)[/tex] and [tex]\(F'(t, 0)\)[/tex].
- Reason: Definition of midpoints.
We need to determine the reasoning for statement 7, which is about [tex]\(\triangle A'B'C'\)[/tex] and its property related to slope and parallelism.
Given choices are:
A. Definition of midpoint
B. Definition of slope
C. Parallel lines have equal slopes.
D. Using point-slope formula
By examining the statements provided:
- Since we know the vertices positions and midpoints, the property we need to justify here is related to the slopes and properties of lines.
- Statement 7 is likely connecting two lines and referring to their slopes, asserting they are equal because they are parallel.
Thus, carefully analyzing the proof step-by-step and understanding what is needed to justify a statement about the slopes of lines being equal as a key property of parallel lines, we conclude:
The reason for statement 7 in the given proof is:
C. Parallel lines have equal slopes.
Here's an overview of the proof provided:
1. Define the vertices of [tex]\(\triangle ABC\)[/tex] at specific points [tex]\(A (x_1, y_1)\)[/tex], [tex]\(B (x_2, y_2)\)[/tex], [tex]\(C(x_3, y_3)\)[/tex].
- Reason: Given.
2. Use rigid transformations to transform [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A'B'C'\)[/tex] such that [tex]\(A'\)[/tex] is at the origin, and [tex]\(\overline{A'C'}\)[/tex] lies along the x-axis.
- Reason: Using rigid transformations, any point can move to any other point, and any line can move to any other line.
3. Any property true for [tex]\(\triangle A'B'C'\)[/tex] is also true for [tex]\(\triangle ABC\)[/tex].
- Reason: Definition of congruence.
4. Use constants [tex]\(r, s, t\)[/tex] to redefine the vertices [tex]\(A'(0,0)\)[/tex], [tex]\(B'(2r, 2s)\)[/tex], [tex]\(C'(2t, 0)\)[/tex].
- Reason: Defining constants.
5. Define midpoints of [tex]\(\overline{A'B'}\)[/tex], [tex]\(\overline{B'C'}\)[/tex], and [tex]\(\overline{A'C'}\)[/tex] as [tex]\(D', E', F'\)[/tex].
- Reason: Defining points.
6. Specify the coordinates for [tex]\(D'\)[/tex] and [tex]\(F'\)[/tex] as [tex]\(D'(r, s)\)[/tex] and [tex]\(F'(t, 0)\)[/tex].
- Reason: Definition of midpoints.
We need to determine the reasoning for statement 7, which is about [tex]\(\triangle A'B'C'\)[/tex] and its property related to slope and parallelism.
Given choices are:
A. Definition of midpoint
B. Definition of slope
C. Parallel lines have equal slopes.
D. Using point-slope formula
By examining the statements provided:
- Since we know the vertices positions and midpoints, the property we need to justify here is related to the slopes and properties of lines.
- Statement 7 is likely connecting two lines and referring to their slopes, asserting they are equal because they are parallel.
Thus, carefully analyzing the proof step-by-step and understanding what is needed to justify a statement about the slopes of lines being equal as a key property of parallel lines, we conclude:
The reason for statement 7 in the given proof is:
C. Parallel lines have equal slopes.
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