Common Mistakes
Traps in Straight Lines
5 mistake patterns students fall for. Each one shows the wrong approach vs the correct approach.
Forgetting absolute value in distance formula
SIGN ERRORDistance is always positive. The formula d = |ax₁+by₁+c|/√(a²+b²) needs the absolute value.
✗ WRONG: d = (ax₁+by₁+c)/√(a²+b²) without absolute value, getting negative distance
✓ RIGHT: Always take |ax₁+by₁+c|. The signed version is useful for checking same/opposite side of a line.
Perpendicular slope: forgetting negative reciprocal
SIGN ERRORIf slope of a line is m, the perpendicular slope is -1/m, not 1/m.
✗ WRONG: Perpendicular to y = 2x has slope 1/2
✓ RIGHT: Perpendicular to y = 2x has slope -1/2. Product of perpendicular slopes = -1.
Confusing centroid, circumcenter, and orthocenter
FORMULAThese are different points. Centroid divides medians 2:1. Orthocenter is intersection of altitudes.
✗ WRONG: Using centroid formula when orthocenter is asked
✓ RIGHT: Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). For orthocenter, find intersection of any two altitudes.
Section formula: internal vs external division
FORMULAInternal division uses + in denominator. External uses - (or negative ratio).
✗ WRONG: Using internal division formula for external division
✓ RIGHT: External division in ratio m:n → use ratio m:(-n) in the formula, or change signs in denominator.
Angle with x-axis vs angle between two lines
FORMULAtan(θ) for angle with x-axis is just the slope. Angle between two lines uses the formula with both slopes.
✗ WRONG: Using angle-between-lines formula when only the inclination is needed
✓ RIGHT: Inclination: θ = arctan(m). Between lines: tanα = |m₁-m₂|/(1+m₁m₂). These are different angles.