Identify the true statement. A Nonparallel lines have the same slope. B Parallel lines have reciprocal slopes of opposite sign. C Perpendicular lines have the same slope. D Perpendicular lines have reciprocal slopes of opposite sign.
The correct answer and explanation is:
Correct Answer: D. Perpendicular lines have reciprocal slopes of opposite sign.
Explanation:
To understand why option D is correct, we need to look at the properties of the slopes of lines in coordinate geometry.
1. What is slope?
The slope of a line measures its steepness and direction. It is calculated as: slope=riserun=y2−y1x2−x1\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{y_2 – y_1}{x_2 – x_1}
2. Perpendicular Lines (Correct Answer D):
Two lines are perpendicular if they intersect at a 90-degree angle. The key property is:
- Their slopes are negative reciprocals of each other.
If one line has slope mm, then the other line has slope −1m-\frac{1}{m}, assuming m≠0m \neq 0.
Example:
- Line A has a slope of 2.
- Line B must then have a slope of −12-\frac{1}{2} to be perpendicular to Line A.
This relationship ensures the lines form a right angle when they intersect.
3. Why the Other Options Are Incorrect:
- A. Nonparallel lines have the same slope
❌ Incorrect. Lines with the same slope are parallel, not nonparallel. - B. Parallel lines have reciprocal slopes of opposite sign
❌ Incorrect. Parallel lines have equal slopes, not reciprocal or opposite. - C. Perpendicular lines have the same slope
❌ Incorrect. If lines have the same slope, they are parallel, not perpendicular.
Conclusion:
The only true statement is D: Perpendicular lines have reciprocal slopes of opposite sign. This is a foundational concept in algebra and geometry and is often used when analyzing or constructing right angles between linear equations on a coordinate plane.