Question
Question 6 (1 point) (04.02 MC) Segment AB is on the line \( y-3=2(x+2) \), and segment CD is on the line \( y-3=-\frac{1}{2}(x+2) \). Which statement proves the relationship of segments \( A B \) and CD ? They are parallel because they have the same slope of 2 They are parallel because they have the same slope of \( -\frac{1}{2} \). They are perpendicular because they have slopes that are opposite reciprocals of 2 and \( -\frac{1}{2} \). They are perpendicular because they have slopes that are opposite reciprocals of -2 and \( \frac{1}{2} \). Ther
Ask by Stuart Rojas. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
Segments AB and CD are perpendicular because their slopes are opposite reciprocals of 2 and -1/2.
Solution
To determine the relationship between segments \( AB \) and \( CD \), we need to analyze the slopes of the lines on which these segments lie.
1. **Identify the slope of line \( AB \)**:
The equation of line \( AB \) is given as:
\[
y - 3 = 2(x + 2)
\]
This is in point-slope form, where the slope \( m_1 = 2 \).
2. **Identify the slope of line \( CD \)**:
The equation of line \( CD \) is given as:
\[
y - 3 = -\frac{1}{2}(x + 2)
\]
This is also in point-slope form, where the slope \( m_2 = -\frac{1}{2} \).
3. **Determine the relationship between the slopes**:
- The slopes are \( m_1 = 2 \) and \( m_2 = -\frac{1}{2} \).
- To check if the lines are perpendicular, we need to see if the product of the slopes equals \(-1\):
\[
m_1 \cdot m_2 = 2 \cdot \left(-\frac{1}{2}\right) = -1
\]
Since the product of the slopes is \(-1\), the lines are perpendicular.
4. **Conclusion**:
The correct statement that proves the relationship of segments \( AB \) and \( CD \) is:
- They are perpendicular because they have slopes that are opposite reciprocals of \( 2 \) and \( -\frac{1}{2} \).
Thus, the answer is:
**They are perpendicular because they have slopes that are opposite reciprocals of 2 and \( -\frac{1}{2} \).**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To determine the relationship between segments AB and CD, you first need to find the slopes of both lines. For line AB, the slope is 2, while for line CD, the slope is \( -\frac{1}{2} \). Since the product of these two slopes is -1 (2 * \( -\frac{1}{2} \) = -1), it proves that segments AB and CD are indeed perpendicular! In geometry, when two lines intersect at a right angle, it forms an awesome L-shape or T-shape, and just like in life, these perpendicular lines signify a perfect complement to each other! If you’re ever lost in a maze, remember: sometimes you need to take those right angles to find your way out!
