Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To solve this problem, we need to determine the coordinates of the image of the triangle [tex]$\triangle ABC$[/tex] after performing two transformations: a [tex]$180^{\circ}$[/tex] clockwise rotation about the origin and a reflection across the line [tex]$y=-x$[/tex]. Let's go through each step in detail:
### Step 1: Rotating [tex]$180^{\circ}$[/tex] clockwise about the origin
When a point [tex]$(x, y)$[/tex] is rotated [tex]$180^{\circ}$[/tex] clockwise about the origin, the new coordinates become [tex]$(-x, -y)$[/tex].
The original coordinates of the vertices are:
- [tex]\( A(-3, 0) \)[/tex]
- [tex]\( B(-2, 3) \)[/tex]
- [tex]\( C(-1, 1) \)[/tex]
Applying the [tex]$180^{\circ}$[/tex] clockwise rotation:
- For vertex [tex]\( A \)[/tex]:
[tex]\[ A' = (-(-3), -(0)) = (3, 0) \][/tex]
- For vertex [tex]\( B \)[/tex]:
[tex]\[ B' = (-(-2), -(3)) = (2, -3) \][/tex]
- For vertex [tex]\( C \)[/tex]:
[tex]\[ C' = (-(-1), -(1)) = (1, -1) \][/tex]
After the [tex]$180^{\circ}$[/tex] rotation, the coordinates of the vertices are:
- [tex]\( A'(3, 0) \)[/tex]
- [tex]\( B'(2, -3) \)[/tex]
- [tex]\( C'(1, -1) \)[/tex]
### Step 2: Reflecting across the line [tex]$y = -x$[/tex]
When a point [tex]$(x, y)$[/tex] is reflected across the line [tex]$y = -x$[/tex], the new coordinates become [tex]$(y, x)$[/tex].
Using the new coordinates from the rotation step:
- For vertex [tex]\( A' \)[/tex]:
[tex]\[ A'' = (0, 3) \][/tex]
- For vertex [tex]\( B' \)[/tex]:
[tex]\[ B'' = (-3, 2) \][/tex]
- For vertex [tex]\( C' \)[/tex]:
[tex]\[ C'' = (-1, 1) \][/tex]
After reflecting across the line [tex]$y = -x$[/tex], the coordinates of the vertices are:
- [tex]\( A''(0, 3) \)[/tex]
- [tex]\( B''(-3, 2) \)[/tex]
- [tex]\( C''(-1, 1) \)[/tex]
Thus, the correct answer is:
### A. [tex]\( A''(0, 3), B''(-3, 2), C''(-1, 1) \)[/tex]
### Step 1: Rotating [tex]$180^{\circ}$[/tex] clockwise about the origin
When a point [tex]$(x, y)$[/tex] is rotated [tex]$180^{\circ}$[/tex] clockwise about the origin, the new coordinates become [tex]$(-x, -y)$[/tex].
The original coordinates of the vertices are:
- [tex]\( A(-3, 0) \)[/tex]
- [tex]\( B(-2, 3) \)[/tex]
- [tex]\( C(-1, 1) \)[/tex]
Applying the [tex]$180^{\circ}$[/tex] clockwise rotation:
- For vertex [tex]\( A \)[/tex]:
[tex]\[ A' = (-(-3), -(0)) = (3, 0) \][/tex]
- For vertex [tex]\( B \)[/tex]:
[tex]\[ B' = (-(-2), -(3)) = (2, -3) \][/tex]
- For vertex [tex]\( C \)[/tex]:
[tex]\[ C' = (-(-1), -(1)) = (1, -1) \][/tex]
After the [tex]$180^{\circ}$[/tex] rotation, the coordinates of the vertices are:
- [tex]\( A'(3, 0) \)[/tex]
- [tex]\( B'(2, -3) \)[/tex]
- [tex]\( C'(1, -1) \)[/tex]
### Step 2: Reflecting across the line [tex]$y = -x$[/tex]
When a point [tex]$(x, y)$[/tex] is reflected across the line [tex]$y = -x$[/tex], the new coordinates become [tex]$(y, x)$[/tex].
Using the new coordinates from the rotation step:
- For vertex [tex]\( A' \)[/tex]:
[tex]\[ A'' = (0, 3) \][/tex]
- For vertex [tex]\( B' \)[/tex]:
[tex]\[ B'' = (-3, 2) \][/tex]
- For vertex [tex]\( C' \)[/tex]:
[tex]\[ C'' = (-1, 1) \][/tex]
After reflecting across the line [tex]$y = -x$[/tex], the coordinates of the vertices are:
- [tex]\( A''(0, 3) \)[/tex]
- [tex]\( B''(-3, 2) \)[/tex]
- [tex]\( C''(-1, 1) \)[/tex]
Thus, the correct answer is:
### A. [tex]\( A''(0, 3), B''(-3, 2), C''(-1, 1) \)[/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. 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.