What is the slope of a line perpendicular to y = 2x + 5?

The slope of a perpendicular line equals -1/2 because perpendicular lines have slopes that are negative reciprocals of each other, and the negative reciprocal of 2 is -1/2, satisfying the geometric condition that the product of their slopes equals -1 (2 × -1/2 = -1).

A. -2

This slope equals the negative of the original slope but not its reciprocal, producing lines that intersect at an angle other than 90 degrees. The error likely originates from recognizing that perpendicular slopes require sign change but omitting the reciprocal relationship executing m⊥ = -m instead of m⊥ = -1/m. Students might recall "negative slope" for perpendicularity but forget the reciprocal component, possibly influenced by reflection over axes where sign changes occur without reciprocal adjustment. Another plausible pathway involves solving 2 × m = -1 → m = -0.5 correctly but then mis-transcribing as -2 through digit transposition. The product 2 × (-2) = -4 ≠ -1 immediately disqualifies this slope for perpendicularity revealing how partial application of the perpendicular slope rule (sign change without reciprocal) produces mathematically invalid results that fail the fundamental orthogonality condition. This distractor effectively identifies learners with fragmented perpendicular slope knowledge who recognize sign inversion is required but lack precision in the reciprocal relationship a transitional deficiency requiring explicit emphasis on the complete negative reciprocal rule with verification through slope product calculation to catch incomplete rule application that compromises geometric accuracy in coordinate geometry contexts.

B. -1/2

This value correctly represents the perpendicular slope through multiple verification pathways demonstrating geometric relationship mastery. Negative reciprocal application: m⊥ = -1/m = -1/2. Orthogonality verification: m₁ × m₂ = 2 × (-1/2) = -1, confirming perpendicularity condition. Graphical interpretation: line y = 2x + 5 rises 2 units per 1 unit right; perpendicular line falls 1 unit per 2 units right, creating right angle intersection. Vector approach: direction vector of original line (1, 2); perpendicular vector (-2, 1) or (2, -1) has slope -1/2 or 1/-2 = -1/2. Angle calculation: tan(θ) = |(m₂ - m₁)/(1 + m₁m₂)|; with m₁m₂ = -1, denominator zero implies θ = 90°. This solution demonstrates comprehensive understanding of perpendicular slope relationships including negative reciprocal rule application, orthogonality condition verification through slope product, graphical slope interpretation, vector perpendicularity translation, and angle formula validation integrated competencies essential for coordinate geometry (line relationships), calculus (normal line determination), physics (orthogonal force components), and computer graphics (collision detection algorithms) where accurate perpendicular slope determination enables spatial relationship analysis, orthogonal decomposition, and geometric constraint satisfaction across contexts requiring precise angular relationships.

C. 1/2

This slope equals the reciprocal of the original slope but lacks the required negative sign, producing lines that intersect at an acute angle rather than 90 degrees. The error likely originates from recognizing the reciprocal relationship but omitting the sign change executing m⊥ = 1/m instead of m⊥ = -1/m. Students might recall "reciprocal slope" for perpendicularity but forget the negative component, possibly influenced by parallel line misconceptions where slopes are identical without sign considerations. Another plausible pathway involves solving 2 × m = 1 (incorrect orthogonality condition) → m = 1/2. The product 2 × (1/2) = 1 ≠ -1 immediately signals non-perpendicularity revealing how partial rule application (reciprocal without sign change) produces mathematically invalid results failing the orthogonality condition. This distractor proves particularly instructive for diagnosing whether students understand the complete negative reciprocal relationship versus isolated components a conceptual gap requiring explicit emphasis on the dual requirements (reciprocal magnitude AND opposite sign) with verification through slope product calculation to prevent incomplete rule application that compromises geometric accuracy in increasingly complex line relationship analyses.

D. 2

This slope equals the original slope itself, producing parallel lines that never intersect rather than perpendicular lines intersecting at 90 degrees. The error likely originates from confusing perpendicular with parallel line relationships recalling that parallel lines share identical slopes but misapplying this property to perpendicular contexts. Students might execute mental search for "special slope relationship" and retrieve parallel slope rule without verifying problem requirements. Another plausible pathway involves solving m⊥ = m = 2 through cognitive overload during multi-concept problems where parallel and perpendicular properties become conflated. The identical slopes immediately signal parallelism rather than perpendicularity revealing fundamental concept confusion between two distinct line relationships requiring explicit differentiation through visual reinforcement contrasting parallel (never meeting) versus perpendicular (meeting at right angles) with slope property summaries: parallel → m₁ = m₂; perpendicular → m₁m₂ = -1. This distractor effectively identifies learners with concept conflation between parallel and perpendicular relationships a foundational deficiency requiring explicit comparative analysis protocols contrasting these relationships through multiple representations (algebraic, graphical, geometric) to build robust conceptual differentiation preventing property misapplication that compromises spatial reasoning accuracy in coordinate geometry contexts.

Conclusion

The perpendicular slope of -1/2 emerges through rigorous application of the negative reciprocal rule recognizing that perpendicular lines require slopes whose product equals -1, verified through multiple independent methods including slope product calculation, graphical interpretation of rise-run relationships, and vector perpendicularity translation. This problem reinforces critical coordinate geometry competencies essential across mathematical domains: understanding the complete negative reciprocal relationship (both reciprocal magnitude AND opposite sign), verifying perpendicularity through slope product condition m₁m₂ = -1, differentiating perpendicular from parallel relationships (identical slopes), and connecting algebraic slope properties to geometric angle relationships. Mastery of these integrated skills proves indispensable for calculus (normal line equations for curves), physics (orthogonal coordinate systems for force resolution), engineering (structural member alignment verification), and computer vision (edge detection algorithms) where accurate perpendicular relationship determination enables spatial analysis, orthogonal decomposition, and geometric constraint satisfaction across contexts requiring precise angular specifications. The distractors strategically target pervasive misconceptions including partial rule application (sign without reciprocal, reciprocal without sign), and concept conflation between parallel and perpendicular relationships highlighting the necessity of explicit complete rule statement with mandatory verification through slope product calculation before accepting any candidate slope, a disciplined habit preventing undetected errors that compromise geometric validity in increasingly complex spatial relationship analyses with real-world implications for design, navigation, and structural integrity assessment.