Which Graph Best Represents A Line Perpendicular To Line K

Finding the Graph that Represents a Line Perpendicular to Line k

Determining which graph represents a line perpendicular to a given line, Line k, requires a fundamental understanding of slope and the relationship between perpendicular lines. This article will guide you through the process, explaining the concepts of slope, perpendicular lines, and how to identify the correct graph. We'll explore various scenarios, provide step-by-step instructions, and address frequently asked questions. By the end, you'll be able to confidently identify the perpendicular line from a set of graphical representations.

Understanding Slope

The slope of a line is a measure of its steepness. It represents the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line. The slope is often denoted by the letter m. A positive slope indicates an upward-sloping line, a negative slope indicates a downward-sloping line, a slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.

The slope can be calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are the coordinates of any two points on the line.

Perpendicular Lines: The Key Relationship

Two lines are perpendicular if they intersect at a right angle (90°). The relationship between the slopes of perpendicular lines is crucial for identifying them graphically. If two lines are perpendicular and neither is vertical, then the product of their slopes is -1. In simpler terms:

m₁ * m₂ = -1

where m₁ is the slope of the first line and m₂ is the slope of the second line. This means that the slope of one line is the negative reciprocal of the slope of the other line. For example, if one line has a slope of 2, its perpendicular line will have a slope of -1/2.

Identifying the Perpendicular Line Graphically: A Step-by-Step Guide

Let's assume we're given the graph of Line k and need to identify the graph representing a line perpendicular to it. Here's a step-by-step approach:

Step 1: Determine the slope of Line k (mₖ).

Examine the graph of Line k. Identify two distinct points on the line and calculate its slope using the formula mentioned above. If Line k is vertical, its slope is undefined; if it's horizontal, its slope is 0.

Step 2: Calculate the slope of the perpendicular line (m⊥).

Use the relationship between the slopes of perpendicular lines:

m⊥ = -1 / mₖ

If mₖ is 0 (horizontal line), m⊥ is undefined (vertical line). If mₖ is undefined (vertical line), m⊥ is 0 (horizontal line).

Step 3: Identify the graph with the calculated slope.

Examine the graphs provided. Look for the line with a slope equal to m⊥. Remember, a positive slope indicates an upward-sloping line, a negative slope indicates a downward-sloping line, a slope of 0 indicates a horizontal line, and an undefined slope indicates a vertical line.

Illustrative Examples

Example 1:

Let's say Line k has a slope of 3 (mₖ = 3). The slope of the line perpendicular to Line k (m⊥) would be:

m⊥ = -1 / 3

You would need to look for a graph representing a line with a slope of -1/3. This line will be downward-sloping and less steep than Line k.

Example 2:

Let's say Line k is a horizontal line (mₖ = 0). The slope of the line perpendicular to Line k (m⊥) would be undefined, meaning it's a vertical line. You would look for a vertical line in the given graphs.

Example 3:

If Line k is a vertical line (undefined slope), the perpendicular line will have a slope of 0, meaning it's a horizontal line.

Advanced Considerations and Special Cases

• Lines with Undefined Slopes: Vertical lines have undefined slopes. A line perpendicular to a vertical line is always a horizontal line (slope = 0).

• Lines with Zero Slopes: Horizontal lines have a slope of zero. A line perpendicular to a horizontal line is always a vertical line (undefined slope).

• Using intercepts: While slope is the primary method, you can also use the x and y intercepts to help visually verify perpendicularity. Perpendicular lines will generally have different intercepts unless they intersect at the origin (0,0).

• Multiple Choice Questions: When presented with multiple choice questions, always calculate the slope of Line k first, then find the negative reciprocal. This will narrow down the options significantly.

Frequently Asked Questions (FAQ)

Q1: What if Line k is represented by an equation, not a graph?

A1: If Line k is given by an equation (e.g., y = 2x + 1), determine its slope from the equation (in this example, the slope is 2). Then, follow steps 2 and 3 above to find the perpendicular line.

Q2: Can two parallel lines also be perpendicular?

A2: No. Parallel lines have the same slope and never intersect. Perpendicular lines intersect at a 90-degree angle and have slopes that are negative reciprocals of each other.

Q3: What if I'm given more than one line to consider?

A3: Calculate the slope of Line k and the negative reciprocal. Then, systematically check the slope of each line given to find the match.

Q4: How do I accurately determine the slope from a graph?

A4: Choose two clearly marked points on the line. Carefully count the vertical and horizontal distances between them. The vertical distance is the rise, and the horizontal distance is the run. The slope is rise/run.

Conclusion

Identifying the graph representing a line perpendicular to a given line requires a solid understanding of slope and the relationship between perpendicular lines. By following the step-by-step guide outlined above, carefully calculating the slope, and applying the negative reciprocal rule, you can confidently determine which graph represents the perpendicular line. Remember to consider special cases, such as vertical and horizontal lines, and use multiple methods to confirm your findings. Mastering this concept is fundamental to understanding linear algebra and its various applications in geometry and beyond. Practice makes perfect, so work through several examples to solidify your understanding.