To prove a line is perpendicular to another, you need to show that the two lines meet at a 90° angle. There are several ways to do this depending on whether you’re working with slopes, coordinates, vectors, or a traditional geometry proof. The method you choose depends on what information you already have.
Using Slopes: The Negative Reciprocal Test
This is the most common method in algebra and coordinate geometry. Two lines are perpendicular if and only if the product of their slopes equals −1. In other words, one slope is the negative reciprocal of the other. If one line has a slope of 3, a perpendicular line has a slope of −1/3. Multiply them together: 3 × (−1/3) = −1.
To apply this, find the slope of each line. If the lines are given as equations, convert them to slope-intercept form (y = mx + b) and identify each slope. If you’re given two points on each line, calculate slope using (y₂ − y₁) / (x₂ − x₁). Then multiply the two slopes. If the result is exactly −1, the lines are perpendicular.
For example, say line A passes through (1, 2) and (3, 6). Its slope is (6 − 2) / (3 − 1) = 2. Line B passes through (0, 5) and (4, 3). Its slope is (3 − 5) / (4 − 0) = −1/2. The product is 2 × (−1/2) = −1, so the lines are perpendicular.
The Special Case: Vertical and Horizontal Lines
The slope product rule breaks down when one line is vertical, because vertical lines have an undefined slope. You can’t multiply something by undefined and get −1. But vertical and horizontal lines are always perpendicular to each other by definition. A horizontal line has a slope of 0, and a vertical line has no defined slope. If you can show one line is horizontal (slope = 0) and the other is vertical (of the form x = some constant), that alone proves perpendicularity.
Using the Converse of the Pythagorean Theorem
When you have three points forming a triangle and you want to prove that two of the sides meet at a right angle, you can use side lengths instead of slopes. The converse of the Pythagorean theorem states: if the square of the longest side equals the sum of the squares of the other two sides, the triangle is a right triangle. The right angle sits at the vertex opposite the longest side, which means the two shorter sides are perpendicular.
Start by using the distance formula to find the length of each side. Label the longest side c and the other two a and b. Then check whether a² + b² = c². If it does, you’ve proven the triangle contains a right angle, and therefore the two sides forming that angle are perpendicular. For instance, a triangle with sides 8, 16, and 8√5 works out to 64 + 256 = 320, and (8√5)² = 320, confirming a right angle.
This method is especially useful in coordinate geometry proofs where you’re given vertices of a shape and need to prove it contains a right angle without calculating slopes.
Using the Dot Product of Vectors
If you’re working in a linear algebra or physics context, the dot product gives a clean test. Two vectors are perpendicular (the formal term is “orthogonal”) if and only if their dot product equals zero.
For two-dimensional vectors, the dot product of (a, b) and (c, d) is ac + bd. So if you have a vector (3, 4) and another vector (−4, 3), their dot product is (3)(−4) + (4)(3) = −12 + 12 = 0. The vectors are perpendicular. This works in three dimensions too: just add the third component’s product to the sum.
To use this for lines, convert each line’s direction into a vector. If a line has slope 2, its direction vector is (1, 2). If another line has slope −1/2, its direction vector is (2, −1). The dot product is (1)(2) + (2)(−1) = 0, confirming perpendicularity. The dot product method is really the slope product rule in different clothing, but it extends naturally to three or more dimensions where “slope” stops being meaningful.
Using Congruent Adjacent Angles
In a traditional two-column or paragraph proof, you can prove two lines are perpendicular by showing they form congruent adjacent angles. When two lines intersect, they create two pairs of adjacent angles that each form a linear pair (meaning they add up to 180°). If you can prove that two adjacent angles in a linear pair are congruent, each must be 90°, and the lines are perpendicular.
This approach appears frequently in geometry proofs involving bisectors or symmetric figures. If a ray bisects a straight angle, the two resulting angles are each 90°, proving the ray is perpendicular to the line forming the straight angle. You may also encounter problems where you prove two angles are congruent using triangle congruence (such as SAS or ASA) and then conclude perpendicularity from there.
Construction With Compass and Straightedge
Sometimes “proving” a line is perpendicular means constructing one and showing why the construction works. The classic method uses a perpendicular bisector. Given a line segment AB, draw two circles of equal radius centered at A and B. These circles intersect at two points on opposite sides of the segment. The line through those two intersection points is the perpendicular bisector of AB.
To construct a perpendicular from an external point P to a line L, draw a circle centered at P large enough to cross L at two points, Q and R. Then construct the perpendicular bisector of QR. Because P is equidistant from Q and R, the bisector passes through P and meets L at a right angle. The proof relies on the fact that any point equidistant from the endpoints of a segment lies on that segment’s perpendicular bisector.
Perpendicularity in Calculus
In calculus, proving perpendicularity comes up when working with tangent and normal lines to a curve. The derivative of a function at a point gives the slope of the tangent line at that point. The normal line, which is perpendicular to the tangent, has a slope equal to the negative reciprocal of the derivative. If f′(x₀) is the derivative at your point, the normal line’s slope is −1/f′(x₀).
So if you need to prove that a particular line is perpendicular to a curve at a given point, compute the derivative at that point and check whether the line’s slope is the negative reciprocal. The underlying principle is identical to the slope product rule, just applied through derivatives rather than static equations.
Choosing the Right Method
- You have equations of two lines: find both slopes and check if their product is −1.
- You have coordinates of three points forming a triangle: use the distance formula and the converse of the Pythagorean theorem.
- You have direction vectors: compute the dot product and check if it equals zero.
- You’re in a geometry proof with angle relationships: show two adjacent angles in a linear pair are congruent, making each 90°.
- You’re working with a curve in calculus: compare the line’s slope to the negative reciprocal of the derivative at the point of intersection.
- One line is vertical and the other is horizontal: state that directly, since vertical and horizontal lines are perpendicular by definition.
In any written proof, the final statement should clearly declare that the lines are perpendicular, using the ⊥ symbol (for example, line AB ⊥ line CD) along with the reason: slopes whose product is −1, a verified right angle, a zero dot product, or congruent linear pair angles.