Question
What composition transformation rule has LMN, L(8,3),M(4,6), and N(5,9) map onto L'M'N',
L' (-10,-3), M' (-6,0), and N' (-7,3)?
a. a reflection over the x-axis and a move to the right by 2 and down by 6
b. a reflection over the y-axis and a move to the left by 6 and down by 2
c. a rotation of 270 degrees clockwise and a move right by 2 and up by 6
d. a reflection over the y-axis and a move to the left by 2 and down by 6
L' (-10,-3), M' (-6,0), and N' (-7,3)?
a. a reflection over the x-axis and a move to the right by 2 and down by 6
b. a reflection over the y-axis and a move to the left by 6 and down by 2
c. a rotation of 270 degrees clockwise and a move right by 2 and up by 6
d. a reflection over the y-axis and a move to the left by 2 and down by 6
Answers
GPT-4o mini
To determine the composition transformation rule that maps the points L(8, 3), M(4, 6), and N(5, 9) onto L'(-10, -3), M'(-6, 0), and N'(-7, 3), we can analyze the changes in coordinates for each point.
1. **Find the change in L:**
- L: (8, 3) to L': (-10, -3)
- Change in x: -10 - 8 = -18
- Change in y: -3 - 3 = -6
2. **Find the change in M:**
- M: (4, 6) to M': (-6, 0)
- Change in x: -6 - 4 = -10
- Change in y: 0 - 6 = -6
3. **Find the change in N:**
- N: (5, 9) to N': (-7, 3)
- Change in x: -7 - 5 = -12
- Change in y: 3 - 9 = -6
Now we can summarize the changes for each point:
- L: change is (-18, -6)
- M: change is (-10, -6)
- N: change is (-12, -6)
From the changes, we can see that while the vertical change (in the y-direction) is consistently -6 for all points, the horizontal changes vary.
Next, we can check which options might be applicable.
Let's closely follow each option and find out if any of them reflects a combined transformation that accomplishes these changes:
- **Option A:** A reflection over the x-axis and a move to the right by 2 and down by 6:
- Reflection over x-axis will change the y-coordinates by negating them.
- Therefore L: (8, 3) to (8, -3), then move to the right by 2 gives (10, -3) and then down by 6 gives (10, -9), which doesn't match L'.
- This option fails.
- **Option B:** A reflection over the y-axis and a move to the left by 6 and down by 2:
- Reflection over y-axis changes L: (8, 3) to (-8, 3).
- Then move left by 6 gives (-14, 3) and down by 2 gives (-14, 1), which is also incorrect.
- This option fails.
- **Option C:** A rotation of 270 degrees clockwise and a move right by 2 and up by 6:
- A 270-degree rotation clockwise changes (x, y) to (y, -x).
- For L(8, 3): (3, -8) → Then right by 2 and up by 6 gives (5, -2), which is not L'.
- So this option fails as well.
- **Option D:** A reflection over the y-axis and a move to the left by 2 and down by 6:
- Reflection over the y-axis changes L(8, 3) to (-8, 3).
- Next, move left by 2 gives (-10, 3) and down by 6 gives (-10, -3), which matches L'.
- Checking M: (4, 6) → (-4, 6) → (-6, 6) → (-6, 0), which matches M'.
- Checking N: (5, 9) → (-5, 9) → (-7, 9) → (-7, 3), which matches N'.
Thus, **Option D** correctly describes the transformation where the reflection over the y-axis and a movement to the left by 2 and down by 6 achieves the required mapping.
**Final Answer: Option D.**
1. **Find the change in L:**
- L: (8, 3) to L': (-10, -3)
- Change in x: -10 - 8 = -18
- Change in y: -3 - 3 = -6
2. **Find the change in M:**
- M: (4, 6) to M': (-6, 0)
- Change in x: -6 - 4 = -10
- Change in y: 0 - 6 = -6
3. **Find the change in N:**
- N: (5, 9) to N': (-7, 3)
- Change in x: -7 - 5 = -12
- Change in y: 3 - 9 = -6
Now we can summarize the changes for each point:
- L: change is (-18, -6)
- M: change is (-10, -6)
- N: change is (-12, -6)
From the changes, we can see that while the vertical change (in the y-direction) is consistently -6 for all points, the horizontal changes vary.
Next, we can check which options might be applicable.
Let's closely follow each option and find out if any of them reflects a combined transformation that accomplishes these changes:
- **Option A:** A reflection over the x-axis and a move to the right by 2 and down by 6:
- Reflection over x-axis will change the y-coordinates by negating them.
- Therefore L: (8, 3) to (8, -3), then move to the right by 2 gives (10, -3) and then down by 6 gives (10, -9), which doesn't match L'.
- This option fails.
- **Option B:** A reflection over the y-axis and a move to the left by 6 and down by 2:
- Reflection over y-axis changes L: (8, 3) to (-8, 3).
- Then move left by 6 gives (-14, 3) and down by 2 gives (-14, 1), which is also incorrect.
- This option fails.
- **Option C:** A rotation of 270 degrees clockwise and a move right by 2 and up by 6:
- A 270-degree rotation clockwise changes (x, y) to (y, -x).
- For L(8, 3): (3, -8) → Then right by 2 and up by 6 gives (5, -2), which is not L'.
- So this option fails as well.
- **Option D:** A reflection over the y-axis and a move to the left by 2 and down by 6:
- Reflection over the y-axis changes L(8, 3) to (-8, 3).
- Next, move left by 2 gives (-10, 3) and down by 6 gives (-10, -3), which matches L'.
- Checking M: (4, 6) → (-4, 6) → (-6, 6) → (-6, 0), which matches M'.
- Checking N: (5, 9) → (-5, 9) → (-7, 9) → (-7, 3), which matches N'.
Thus, **Option D** correctly describes the transformation where the reflection over the y-axis and a movement to the left by 2 and down by 6 achieves the required mapping.
**Final Answer: Option D.**