Difficulty: Easy
Correct Answer: -5
Explanation:
Introduction:
This coordinate geometry question checks your understanding of slopes of lines and the relationship between the slopes of perpendicular lines. Once you find the slope of the given line, you can use the negative reciprocal to obtain the slope of any line perpendicular to it.
Given Data / Assumptions:
Concept / Approach:
The slope m of a line passing through (x₁, y₁) and (x₂, y₂) is: m = (y₂ − y₁) / (x₂ − x₁). If two lines are perpendicular and neither is vertical, then the product of their slopes is −1. Thus, if one line has slope m, the perpendicular line has slope m⊥ = −1 / m. This negative reciprocal relationship is the key to the solution.
Step-by-Step Solution:
First find the slope of the given line through (8, 2) and (3, 1). m = (1 − 2) / (3 − 8) = (−1) / (−5) = 1/5. So the slope of the given line is 1/5. For a line perpendicular to this, we need a slope m⊥ such that m * m⊥ = −1. So (1/5) * m⊥ = −1. Therefore m⊥ = −1 / (1/5) = −5.
Verification / Alternative check:
You can verify by checking the product: (1/5) * (−5) = −1, which confirms perpendicularity. This is the standard condition for two non vertical lines to be perpendicular in the plane.
Why Other Options Are Wrong:
The slope 5 has product 5 * (1/5) = 1, indicating parallelism in a different sense, not perpendicularity. The slopes 1/5 and −1/5 are either the original slope or a line that is not perpendicular because their product with 1/5 is not −1. Only −5 satisfies the negative reciprocal condition.
Common Pitfalls:
A common error is to change only the sign or only invert the slope instead of doing both. Another mistake is miscalculating the original slope by mixing up x and y differences. Always compute the initial slope carefully, then take its negative reciprocal for a perpendicular line.
Final Answer:
The slope of the line perpendicular to the given line is −5.
Discussion & Comments