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Sagot :
Let's walk through the question step-by-step to understand how we reach the reason for statement 7.
1. Define the vertices of [tex]\(\triangle ABC\)[/tex]:
- We have three unique points: [tex]\(A(x_1, y_1)\)[/tex], [tex]\(B(x_2, y_2)\)[/tex], and [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]:
- Move [tex]\(A'\)[/tex] to the origin [tex]\((0,0)\)[/tex] and place [tex]\(\overline{A'C'}\)[/tex] on the x-axis.
- Reason: In the coordinate plane, any point can be moved to any other point and any line can be moved to any other line using rigid transformations.
3. Property consistency between [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A'B'C'\)[/tex]:
- If a property is true for [tex]\(\triangle A'B'C'\)[/tex], it is also true for [tex]\(\triangle ABC\)[/tex].
- Reason: Definition of congruence.
4. Vertices of [tex]\(\triangle A'B'C'\)[/tex]:
- Let [tex]\(r, s, t\)[/tex] be real numbers such that [tex]\(A'(0, 0)\)[/tex], [tex]\(B'(2r, 2s)\)[/tex], and [tex]\(C'(2t, 0)\)[/tex].
- Reason: Defining constants.
5. Midpoints of segments [tex]\(\overline{A'B'}\)[/tex], [tex]\(\overline{B'C'}\)[/tex], and [tex]\(\overline{A'C'}\)[/tex]:
- Let [tex]\(D'\)[/tex], [tex]\(E'\)[/tex], and [tex]\(F'\)[/tex] be the midpoints of [tex]\(\overline{A'B'}\)[/tex], [tex]\(\overline{B'C'}\)[/tex], and [tex]\(\overline{A'C'}\)[/tex], respectively.
- Reason: Defining points.
From these steps, we notice that statement 5 involves defining the midpoints, which directly involves the use of the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\(M\)[/tex] of a line segment connecting points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are:
[tex]\[M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\][/tex]
Given that the setup and subsequent transformations place emphasis on the placement and properties of these midpoints in relation to their segments, the most relevant reason for statement 7 pertains to ensuring midpoint properties.
Thus, the reason for statement 7 in the given proof is indeed:
A. definition of midpoint
1. Define the vertices of [tex]\(\triangle ABC\)[/tex]:
- We have three unique points: [tex]\(A(x_1, y_1)\)[/tex], [tex]\(B(x_2, y_2)\)[/tex], and [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]:
- Move [tex]\(A'\)[/tex] to the origin [tex]\((0,0)\)[/tex] and place [tex]\(\overline{A'C'}\)[/tex] on the x-axis.
- Reason: In the coordinate plane, any point can be moved to any other point and any line can be moved to any other line using rigid transformations.
3. Property consistency between [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle A'B'C'\)[/tex]:
- If a property is true for [tex]\(\triangle A'B'C'\)[/tex], it is also true for [tex]\(\triangle ABC\)[/tex].
- Reason: Definition of congruence.
4. Vertices of [tex]\(\triangle A'B'C'\)[/tex]:
- Let [tex]\(r, s, t\)[/tex] be real numbers such that [tex]\(A'(0, 0)\)[/tex], [tex]\(B'(2r, 2s)\)[/tex], and [tex]\(C'(2t, 0)\)[/tex].
- Reason: Defining constants.
5. Midpoints of segments [tex]\(\overline{A'B'}\)[/tex], [tex]\(\overline{B'C'}\)[/tex], and [tex]\(\overline{A'C'}\)[/tex]:
- Let [tex]\(D'\)[/tex], [tex]\(E'\)[/tex], and [tex]\(F'\)[/tex] be the midpoints of [tex]\(\overline{A'B'}\)[/tex], [tex]\(\overline{B'C'}\)[/tex], and [tex]\(\overline{A'C'}\)[/tex], respectively.
- Reason: Defining points.
From these steps, we notice that statement 5 involves defining the midpoints, which directly involves the use of the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\(M\)[/tex] of a line segment connecting points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are:
[tex]\[M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\][/tex]
Given that the setup and subsequent transformations place emphasis on the placement and properties of these midpoints in relation to their segments, the most relevant reason for statement 7 pertains to ensuring midpoint properties.
Thus, the reason for statement 7 in the given proof is indeed:
A. definition of midpoint
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