A line has equation y = (2/3)x + 5. What is the slope of a line perpendicular to it?
A2/3
B3/2
C-2/3
D-3/2
Perpendicular slopes are negative reciprocals: flip the fraction AND change the sign. Starting with 2/3, flip to get 3/2, then negate to get -3/2. Option B (3/2) is the most common wrong answer — it's the reciprocal but forgets the sign change. Option C (-2/3) negates but forgets to flip. Both partial operations produce a slope at some angle other than 90°. The product check confirms: (2/3) × (-3/2) = -1, as required for perpendicular lines.
Question 2 Multiple Choice
A student writes y = -4x + 3 as the equation of a line parallel to y = 4x - 1 passing through (0, 3), reasoning that parallel lines must have opposite slopes. What is the error?
AThe student is correct — parallel lines have slopes that negate each other to balance
BParallel lines require slopes that are reciprocals, not negatives
CParallel lines have equal slopes; the correct equation is y = 4x + 3
DThe line through (0, 3) should have slope 0 because it crosses the y-axis
Parallel lines have exactly equal slopes — they rise at the same rate and therefore never intersect. The only difference between parallel lines is their y-intercepts. Here, the original slope is 4, so any parallel line also has slope 4. The student confused the rule for perpendicular lines (opposite sign) with the rule for parallel lines (same slope). The correct parallel line through (0, 3) is y = 4x + 3.
Question 3 True / False
If two lines are perpendicular, their slopes are reciprocals of each other — for example, 3/4 and 4/3.
TTrue
FFalse
Answer: False
Perpendicular slopes are NEGATIVE reciprocals, not just reciprocals. A line with slope 3/4 has a perpendicular slope of -4/3, not 4/3. Forgetting the sign change is the most common error with perpendicular slopes — it produces a line at a different angle, not a right angle. You can verify: (3/4) × (-4/3) = -1, confirming perpendicularity. (3/4) × (4/3) = 1, which confirms the lines would form equal angles with the x-axis (like a reflection), not a right angle.
Question 4 True / False
A vertical line (undefined slope) and a horizontal line (slope = 0) are perpendicular, even though the formula m₁ × m₂ = -1 cannot be applied to this case.
TTrue
FFalse
Answer: True
The product formula breaks down when one slope is undefined (vertical line), since you cannot multiply by undefined. However, the geometric relationship is unambiguous: vertical and horizontal lines meet at a perfect right angle. This is a special case that must be recognized on its own terms rather than by formula. The underlying geometric truth holds even when the algebraic shortcut fails — geometry doesn't stop working just because a formula has a gap.
Question 5 Short Answer
Why does finding a perpendicular slope require BOTH flipping the fraction AND changing the sign, rather than just one of those operations?
Think about your answer, then reveal below.
Model answer: When a line is rotated exactly 90 degrees, two independent geometric changes happen simultaneously: the roles of rise and run swap (the new run is the old rise, and vice versa), which flips the fraction; and the direction of travel reverses (what went upward now goes downward), which changes the sign. Each operation corresponds to a distinct geometric transformation. Doing only the flip produces a slope with the same sign — a reflection across a diagonal, not a right angle. Doing only the sign change produces a slope that mirrors across the x-axis, also not a right angle. Both together produce the 90° rotation.
This is why the negative reciprocal relationship can be understood rather than memorized: there are two independent things that happen to a line when you rotate it 90°, and each has its own algebraic consequence. The product rule (m₁ × m₂ = -1) encodes both requirements simultaneously, which is why it works as a verification tool.