What is the slope of a line that is perpendicular to the line passing through the points (-5, 1) and (-2, 0)?

Introduction / Context:
In coordinate geometry, slopes help describe the orientation of lines. Two lines are perpendicular if the angle between them is 90°, and there is a specific relationship between their slopes. This question tests your ability to compute the slope of a line from two points and then use the negative reciprocal rule to find the slope of a perpendicular line.

Given Data / Assumptions:

• A line passes through points A(-5, 1) and B(-2, 0).
• We need the slope of any line that is perpendicular to this line.
• The coordinate plane is standard with x horizontal and y vertical.

Concept / Approach:
First, find the slope m of the given line using the formula:
m = (y2 - y1) / (x2 - x1) If a line has slope m, then any line perpendicular to it has slope m_perp given by:
m_perp = -1 / m This is known as the negative reciprocal relationship between slopes of perpendicular lines (provided neither line is vertical or horizontal in a conflicting way).

Step-by-Step Solution:
Step 1: Label the points A(-5, 1) and B(-2, 0). Step 2: Compute the slope m of line AB using m = (y2 - y1) / (x2 - x1). Step 3: Substitute the coordinates: m = (0 - 1) / (-2 - (-5)). Step 4: Simplify the numerator: 0 - 1 = -1. Step 5: Simplify the denominator: -2 - (-5) = -2 + 5 = 3. Step 6: Therefore, m = -1 / 3. Step 7: The slope of a perpendicular line is the negative reciprocal of -1 / 3, so m_perp = -1 / (-1 / 3) = 3.

Verification / Alternative check:
If two slopes m1 and m2 correspond to perpendicular lines, then their product should be -1. Here, m1 = -1/3 and m2 = 3. The product is (-1/3) * 3 = -1, which satisfies the perpendicular condition exactly. This confirms that the slope 3 is correct for a perpendicular line.

Why Other Options Are Wrong:

• -3: This would make the product of slopes (-1/3) * (-3) = 1, indicating parallel but not perpendicular lines.
• -1/3: This is the slope of the original line, so it gives parallel lines, not perpendicular ones.
• 1/3: This gives a product of slopes (-1/3) * (1/3) = -1/9, which does not indicate perpendicularity.
• 0: This slope would give a horizontal line, which is perpendicular only to vertical lines, not to the given line with slope -1/3.

Common Pitfalls:
Students frequently confuse "negative reciprocal" with simply "negative" or merely "reciprocal". Another error is making sign mistakes while subtracting coordinates. Always calculate the original slope carefully, then take the negative reciprocal accurately to find the slope of a line perpendicular to it.

Final Answer:
Therefore, the slope of a line perpendicular to the line through (-5, 1) and (-2, 0) is 3.