Arcs and Chords

Geometry · Axiom Academy

Understanding arcs, chords, and their relationships 1. Chords and the Arcs They Cut Drop two points on a circle and join them with a straight segment — that's a chord . The same two points split the circumference into two pieces; each piece is an arc . So every chord comes paired with an arc it "subtends." A chord's two endpoints, say A and B , give two arcs: a shorter one and a longer one. We classify them by how many degrees they span. Less than 180°. Named with two letters: . Greater than 180°. Named with three letters: . Here is the key fact. In the same circle (or congruent circles), two chords of equal length cut off arcs of equal measure — and the converse holds too. Equal chords ⟺ equal arcs (same or congruent circles) Four theorems do most of the work in chord problems. The third one — the perpendicular from the center — is the workhorse, and we animate it because it creates a right triangle you can solve. In the same circle (or congruent circles), two chords are congruent if and only if their corresponding arcs are congruent. If a line through the center is perpendicular to a chord, it bisects the chord and its arc. In the same circle, two chords are congruent if and only if they are equidistant from the center. The diameter is the longest chord in any circle; every other chord is shorter. 5. Example: Finding an Arc Measure Use the congruent-chords theorem directly. In circle O , chord AB = chord CD . If arc AB measures 70°, what is the measure of arc CD ?

This is the written version of the interactive lesson above. See the full Geometry course.