Question
Question
What composition transformation rule has applied to have △LMN
map onto △L′M′N′
, if the coordinates are:
L(8,3)
, M(4,6)
, and N(5,9)
L′(−10,−3)
, M′(−6,0)
, and N′(−7,3)
?
(1 point)
Responses
a reflection across the x-axis and a move to the right by 2 and down by 6
a reflection across the x-axis and a move to the right by 2 and down by 6
a rotation of 270 degrees clockwise and a move right by 2 and up by 6
a rotation of 270 degrees clockwise and a move right by 2 and up by 6
a reflection over the y-axis and a move to the left by 6 and down by 2
a reflection over the y-axis and a move to the left by 6 and down by 2
a reflection over the y-axis and a move to the left by 2 and down by 6
a reflection over the y-axis and a move to the left by 2 and down by 6
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What composition transformation rule has applied to have △LMN
map onto △L′M′N′
, if the coordinates are:
L(8,3)
, M(4,6)
, and N(5,9)
L′(−10,−3)
, M′(−6,0)
, and N′(−7,3)
?
(1 point)
Responses
a reflection across the x-axis and a move to the right by 2 and down by 6
a reflection across the x-axis and a move to the right by 2 and down by 6
a rotation of 270 degrees clockwise and a move right by 2 and up by 6
a rotation of 270 degrees clockwise and a move right by 2 and up by 6
a reflection over the y-axis and a move to the left by 6 and down by 2
a reflection over the y-axis and a move to the left by 6 and down by 2
a reflection over the y-axis and a move to the left by 2 and down by 6
a reflection over the y-axis and a move to the left by 2 and down by 6
Skip to navigation
Answers
Answered by
GPT-4o mini
To determine the composition transformation that maps triangle \( \triangle LMN \) onto triangle \( \triangle L'M'N' \), we start by analyzing the coordinates of the points.
The original coordinates of the points are:
- \( L(8, 3) \)
- \( M(4, 6) \)
- \( N(5, 9) \)
The transformed coordinates of the points are:
- \( L'(-10, -3) \)
- \( M'(-6, 0) \)
- \( N'(-7, 3) \)
### Step 1: Analyze the transformation
First, we will check for a reflection across the x-axis, y-axis, or a rotation:
1. **Reflection across the x-axis** changes \( (x, y) \) to \( (x, -y) \).
Applying this transformation:
- \( L(8, 3) \rightarrow L(8, -3) \)
- \( M(4, 6) \rightarrow M(4, -6) \)
- \( N(5, 9) \rightarrow N(5, -9) \)
These do not match \( L'(-10, -3) \), \( M'(-6, 0) \), \( N'(-7, 3) \).
2. **Reflection across the y-axis** changes \( (x, y) \) to \( (-x, y) \).
Applying this transformation:
- \( L(8, 3) \rightarrow L'(-8, 3) \)
- \( M(4, 6) \rightarrow M'(-4, 6) \)
- \( N(5, 9) \rightarrow N'(-5, 9) \)
Again, these do not match \( L'(-10, -3) \), \( M'(-6, 0) \), \( N'(-7, 3) \).
3. **Rotation** and translations need to be applied since reflection alone doesn't work.
### Step 2: Calculate the translation after checking for lines
To find distances in x and y more accurately, we can find the difference in both coordinates.
Let's look at how much \( L \) moves:
- From \( L(8, 3) \) to \( L'(-10, -3) \)
- Change in x: \( -10 - 8 = -18 \)
- Change in y: \( -3 - 3 = -6 \)
Similarly, looking at how \( M \) moves:
- From \( M(4, 6) \) to \( M'(-6, 0) \)
- Change in x: \( -6 - 4 = -10 \)
- Change in y: \( 0 - 6 = -6 \)
Finally, how \( N \) moves:
- From \( N(5, 9) \) to \( N'(-7, 3) \)
- Change in x: \( -7 - 5 = -12 \)
- Change in y: \( 3 - 9 = -6 \)
### Step 3: Finding the consistent transformation
The consistent change in the y-coordinate \( (-6) \) across all points shows a downward movement, while the x-coordinates differ significantly.
### Conclusion
The transformation mapping is a **reflection across the y-axis** (due to all x-coordinates turning negative) followed by a translation that matches:
- A move to the left by 2 and down by 6 (after observing that translations are consistent placements along negative lines).
Thus, the answer is:
**a reflection over the y-axis and a move to the left by 2 and down by 6.**
The original coordinates of the points are:
- \( L(8, 3) \)
- \( M(4, 6) \)
- \( N(5, 9) \)
The transformed coordinates of the points are:
- \( L'(-10, -3) \)
- \( M'(-6, 0) \)
- \( N'(-7, 3) \)
### Step 1: Analyze the transformation
First, we will check for a reflection across the x-axis, y-axis, or a rotation:
1. **Reflection across the x-axis** changes \( (x, y) \) to \( (x, -y) \).
Applying this transformation:
- \( L(8, 3) \rightarrow L(8, -3) \)
- \( M(4, 6) \rightarrow M(4, -6) \)
- \( N(5, 9) \rightarrow N(5, -9) \)
These do not match \( L'(-10, -3) \), \( M'(-6, 0) \), \( N'(-7, 3) \).
2. **Reflection across the y-axis** changes \( (x, y) \) to \( (-x, y) \).
Applying this transformation:
- \( L(8, 3) \rightarrow L'(-8, 3) \)
- \( M(4, 6) \rightarrow M'(-4, 6) \)
- \( N(5, 9) \rightarrow N'(-5, 9) \)
Again, these do not match \( L'(-10, -3) \), \( M'(-6, 0) \), \( N'(-7, 3) \).
3. **Rotation** and translations need to be applied since reflection alone doesn't work.
### Step 2: Calculate the translation after checking for lines
To find distances in x and y more accurately, we can find the difference in both coordinates.
Let's look at how much \( L \) moves:
- From \( L(8, 3) \) to \( L'(-10, -3) \)
- Change in x: \( -10 - 8 = -18 \)
- Change in y: \( -3 - 3 = -6 \)
Similarly, looking at how \( M \) moves:
- From \( M(4, 6) \) to \( M'(-6, 0) \)
- Change in x: \( -6 - 4 = -10 \)
- Change in y: \( 0 - 6 = -6 \)
Finally, how \( N \) moves:
- From \( N(5, 9) \) to \( N'(-7, 3) \)
- Change in x: \( -7 - 5 = -12 \)
- Change in y: \( 3 - 9 = -6 \)
### Step 3: Finding the consistent transformation
The consistent change in the y-coordinate \( (-6) \) across all points shows a downward movement, while the x-coordinates differ significantly.
### Conclusion
The transformation mapping is a **reflection across the y-axis** (due to all x-coordinates turning negative) followed by a translation that matches:
- A move to the left by 2 and down by 6 (after observing that translations are consistent placements along negative lines).
Thus, the answer is:
**a reflection over the y-axis and a move to the left by 2 and down by 6.**