The upside-down T in geometry is the perpendicular symbol (⊥), and it means two lines intersect at exactly 90 degrees. When you see something like line AB ⊥ line CD, it’s a shorthand way of saying those two lines form a right angle where they cross.
What the Symbol Tells You
The ⊥ symbol applies to lines, line segments, and rays. Whenever two of these meet at a perfect right angle, they’re perpendicular, and the upside-down T is the standard notation for that relationship. You’ll see it written between two lines (AB ⊥ CD) or used as a label on diagrams where a small square marks the 90-degree corner.
One detail worth knowing: when two straight lines cross at 90 degrees, they actually create four right angles at the intersection point. If any one of the four angles measures 90 degrees, the other three automatically do too. So perpendicularity isn’t about just one corner of the intersection. It defines the entire crossing.
Perpendicular Lines on a Coordinate Plane
In coordinate geometry, perpendicularity has a clean mathematical test. If one line has a slope of m₁ and another has a slope of m₂, the two lines are perpendicular when m₁ × m₂ = −1. In plain terms, the slope of a perpendicular line is the negative reciprocal of the original. A line with a slope of 2 is perpendicular to a line with a slope of −1/2, because 2 × (−1/2) = −1.
This gives you a quick way to check whether two lines are perpendicular without graphing them. Just multiply their slopes. If you get −1, the ⊥ symbol applies.
Perpendicularity in Three Dimensions
The same symbol extends into 3D geometry, where it describes the relationship between a line and a plane or between two planes. A line that hits a flat surface at a 90-degree angle is called a normal line, and it’s perpendicular to every line on that surface passing through the point of contact. This concept shows up constantly in physics and engineering whenever you need to describe a direction pointing straight out from a surface.
In higher math, particularly linear algebra, the ⊥ symbol means “orthogonal,” which is just a generalized version of perpendicular. Two vectors are orthogonal when the angle between them is 90 degrees, which happens when their dot product equals zero. Entire subspaces can be orthogonal to each other, meaning every vector in one subspace is at a right angle to every vector in the other. The row space of a matrix, for example, is orthogonal to its nullspace.
How to Construct a Perpendicular Line
If you’re in a geometry class, you’ll likely need to construct perpendicular lines with a compass and straightedge. Here’s how to build a line perpendicular to a given line through a point that sits on that line:
- Step 1: Place your compass on the point and draw an arc that crosses the line on both sides.
- Step 2: Widen the compass. This is important, because if you keep the same width, the next arcs won’t intersect properly.
- Step 3: Place the compass where the first arc crossed the line on one side and draw a small arc above (or below) the line.
- Step 4: Without changing the compass width, repeat from the other crossing point. The two small arcs should intersect.
- Step 5: Draw a straight line from that intersection point down to your original point. That new line is perpendicular to the original.
Other Meanings of the ⊥ Symbol
Outside geometry, the same upside-down T carries different meanings depending on the field. In logic, it represents “falsum” or “absurdity,” essentially a statement that’s always false. Its Unicode name is actually “up tack” (U+22A5), and in LaTeX it’s typed as \perp or \bot. But in any geometry or spatial math context, it reliably means one thing: a 90-degree angle between two objects.