Technique Encyclopedia
Auxiliary Lines
What it is
An auxiliary line is any segment you add to a geometry figure that wasn't part of the original problem — connecting two existing points, dropping a perpendicular, or extending a side until it meets another line. Many geometry problems look stuck simply because the figure as given doesn't yet contain a usable triangle or right angle; the right auxiliary line creates one.
Signals that suggest using it
- A geometry problem's given figure has no obvious right triangle or usable relationship between the given quantities.
- The problem involves a shape (square, regular polygon) whose area or a key length is more natural to find via a triangle within it.
- A tangent line to a circle is involved — drawing the radius to the point of tangency always creates a right angle.
When it's effective
Whenever a natural connection, altitude, or extension turns an intractable figure into one or more right triangles with computable legs — the Pythagorean theorem or basic trigonometry usually finishes the problem from there.
When it's not effective
Adding lines without a clear target (what right angle or known-length triangle the new line is meant to create) rarely helps — every auxiliary line should be added with a specific goal in mind, not as a hopeful guess.
Simple example
Problem
Find the area of an equilateral triangle with side length 4.
Solution
Draw the altitude from one vertex to the midpoint of the opposite side. This splits the triangle into two 30-60-90 triangles, giving an altitude of 4·(√3/2) = 2√3. Area = ½ · 4 · 2√3 = 4√3.
AMC-style example
Problem
A square has diagonal length 10. Find its area.
Solution
Draw the diagonal, splitting the square into two 45-45-90 triangles. If the side length is s, then s√2 = 10, so s² = 50. The area is exactly s², which is 50.