Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
