Arcs and chords

Learning Objectives Identify and define chords, arcs (minor, major, semicircle), and their related properties. Apply the theorem that congruent chords correspond to congruent arcs, and vice versa. Use the properties of a radius or diameter perpendicular to a chord to find unknown lengths and arc measures. Apply the theorem that congruent chords are equidistant from the center of the circle to solve problems. Solve algebraic and geometric problems by synthesizing multiple theorems about arcs and chords. Construct simple proofs involving the relationships between arcs and chords. Ever wondered how archaeologists can figure out the full size of a broken circular plate from just a small piece? 🏺 It all comes down to the relationship between arcs and chords! This tutorial explo...

TermDefinitionExample ChordA line segment whose endpoints both lie on the circle.In a pizza, if you make a straight cut from one point on the crust to another without passing through the center, the line of that cut is a chord. ArcA portion of the circumference of a circle.The crust of a single slice of pizza represents an arc of the entire pizza crust. Minor ArcAn arc whose measure is less than 180°. It is the shortest path between two points on the circumference and is named using two letters.If points A and B are on a circle, the shorter arc connecting them is minor arc AB, written as \(\overset{\frown}{AB}\). Major ArcAn arc whose measure is greater than 180°. It is the longer path between two points on the circumference and is named using three letters.If points A, B, and C are on a...

Congruent Chords and Arcs Theorem In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. \(\overset{\frown}{AB} \cong \overset{\frown}{CD} \iff \overline{AB} \cong \overline{CD}\) This is a two-way relationship. If you know two chords have the same length, you know their corresponding minor arcs have the same degree measure. If you know two minor arcs have the same degree measure, you know their chords are equal in length. Perpendicular Diameter/Radius to a Chord Theorem If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc. If \(\overline{CD} \perp \overline{AB}\), then \(\overline{AE} \cong \overline{EB}\) and \(\overset{\frown}{AD} \cong \overset{\frow...