Question
Points Upper A prime and Upper B prime are images of points A and B after a 270degrees rotation about the origin. If the slope of ModifyingAbove AB with left right arrow is minus3, what is the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow? Explain.
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Part 1
Select the correct choice below and fill in the answer box to complete your choice.
A.
A rotation of 270degrees would result in a line parallel to ModifyingAbove AB with left right arrow. Since the slopes of parallel lines are equal, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
B.
A rotation of 270degrees would result in a line parallel to ModifyingAbove AB with left right arrow. Since the slopes of parallel lines are opposite reciprocals, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
C.
A rotation of 270degrees would result in a line perpendicular to ModifyingAbove AB with left right arrow. Since the slopes of perpendicular lines are opposite reciprocals, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
D.
A rotation of 270degrees would result in a line perpendicular to ModifyingAbove AB with left right arrow. Since the slopes of perpendicular lines are equal, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
Question content area bottom
Part 1
Select the correct choice below and fill in the answer box to complete your choice.
A.
A rotation of 270degrees would result in a line parallel to ModifyingAbove AB with left right arrow. Since the slopes of parallel lines are equal, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
B.
A rotation of 270degrees would result in a line parallel to ModifyingAbove AB with left right arrow. Since the slopes of parallel lines are opposite reciprocals, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
C.
A rotation of 270degrees would result in a line perpendicular to ModifyingAbove AB with left right arrow. Since the slopes of perpendicular lines are opposite reciprocals, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
D.
A rotation of 270degrees would result in a line perpendicular to ModifyingAbove AB with left right arrow. Since the slopes of perpendicular lines are equal, the slope of ModifyingAbove Upper A prime Upper B prime with left right arrow is
enter your response here.
Answers
GPT-4o mini
To determine the slope of line segment \( \overline{A'B'} \) after a 270-degree rotation of points \( A \) and \( B \) around the origin, we first need to understand the effects of such a rotation on slopes.
1. **Understanding the rotation:**
- A rotation of 270 degrees about the origin can be visualized as a 90-degree rotation counterclockwise or a 270-degree clockwise.
- For a line that has a slope \( m \), after a 270-degree rotation, the slope of the new line \( \overline{A'B'} \) would be the negative reciprocal of the original slope \( m \).
2. **Given:**
- The slope of line segment \( \overline{AB} \) is -3.
3. **Finding the new slope:**
- To find the slope of \( \overline{A'B'} \), we take the negative reciprocal of -3:
\[
\text{slope of } \overline{A'B'} = -\frac{1}{(-3)} = \frac{1}{3}.
\]
Now let's analyze the possible answer choices:
A. Incorrect. This implies that \( \overline{A'B'} \) is parallel to \( \overline{AB} \) and would have the same slope.
B. Incorrect. This option suggests that parallel lines have opposite reciprocals, which is false.
C. Correct. As discussed, a 270-degree rotation results in a line perpendicular to \( \overline{AB} \), and the slopes of perpendicular lines are opposite reciprocals.
D. Incorrect. This incorrectly states that the slopes of perpendicular lines are equal.
Thus, the correct choice is **C**.
**Final answer in the box:**
\[
\frac{1}{3}
\]
1. **Understanding the rotation:**
- A rotation of 270 degrees about the origin can be visualized as a 90-degree rotation counterclockwise or a 270-degree clockwise.
- For a line that has a slope \( m \), after a 270-degree rotation, the slope of the new line \( \overline{A'B'} \) would be the negative reciprocal of the original slope \( m \).
2. **Given:**
- The slope of line segment \( \overline{AB} \) is -3.
3. **Finding the new slope:**
- To find the slope of \( \overline{A'B'} \), we take the negative reciprocal of -3:
\[
\text{slope of } \overline{A'B'} = -\frac{1}{(-3)} = \frac{1}{3}.
\]
Now let's analyze the possible answer choices:
A. Incorrect. This implies that \( \overline{A'B'} \) is parallel to \( \overline{AB} \) and would have the same slope.
B. Incorrect. This option suggests that parallel lines have opposite reciprocals, which is false.
C. Correct. As discussed, a 270-degree rotation results in a line perpendicular to \( \overline{AB} \), and the slopes of perpendicular lines are opposite reciprocals.
D. Incorrect. This incorrectly states that the slopes of perpendicular lines are equal.
Thus, the correct choice is **C**.
**Final answer in the box:**
\[
\frac{1}{3}
\]