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MathChapter 19: Circles

Notes

Arcs, Angles, and Tangents Notes

Name circle parts, classify and combine arcs, and use the two essential tangent theorems in SAT geometry.

Practice this topic
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Circle questions become manageable when every line is classified first. Decide whether a segment is a radius or chord and whether a line is a secant or tangent before writing an equation.

Circle anatomy

Circle anatomyCircle O identifies a radius, chord, secant, tangent, and point of tangency using distinct segments and lines.
Circle anatomyCircle O identifies a radius, chord, secant, tangent, and point of tangency using distinct segments and lines.secanttangentchordOABCD
Circle anatomy
Circle terminology
ObjectDefining featureSAT consequence
RadiusCenter to a point on the circleAll radii of one circle are congruent.
ChordBoth endpoints on the circleA diameter is the longest chord.
SecantA line meeting the circle twiceIt contains a chord.
TangentA line meeting the circle onceIt is perpendicular to the radius at contact.

Central angles and arc measures

Minor, major, and semicircular arcsTwo radii form an 80-degree central angle. The short intercepted arc is minor, the remaining route is major, and a diameter would separate two semicircles.
Minor, major, and semicircular arcsTwo radii form an 80-degree central angle. The short intercepted arc is minor, the remaining route is major, and a diameter would separate two semicircles.80°OAB
Minor, major, and semicircular arcs
Arc classification
ArcMeasureNaming cue
MinorLess than \(180^\circ\)Usually two endpoint letters
SemicircleExactly \(180^\circ\)Endpoints form a diameter
MajorGreater than \(180^\circ\)Use three letters to show the route
Tangent right-triangle relationship
\[PT^2=PO^2-r^2\]

Radius OT is perpendicular to tangent PT, so OP is the hypotenuse of right triangle OPT.

Arc additionAdjacent arcs measure 70 degrees and 45 degrees, so their combined arc measures 115 degrees.
Arc additionAdjacent arcs measure 70 degrees and 45 degrees, so their combined arc measures 115 degrees.70°45°OABC
Arc addition

Tangent theorems

Radius perpendicular to a tangentRadius OT meets the tangent line at T at a marked right angle.
Radius perpendicular to a tangentRadius OT meets the tangent line at T at a marked right angle.rTO
Radius perpendicular to a tangent
Tangents from one exterior pointTwo tangent segments from exterior point P meet the circle at A and B and have equal length.
Tangents from one exterior pointTwo tangent segments from exterior point P meet the circle at A and B and have equal length.xxPOAB
Tangents from one exterior point
Worked example

Tangent triangle

From P, segment PT is tangent to a circle with center O. If \(OP=13\) and radius \(OT=5\), find PT.

  1. Recognize the right angle

    \(OT\perp PT\), so triangle OPT is right at T.

  2. Use Pythagorean Theorem

    \(PT^2+5^2=13^2\), so \(PT^2=144\).

\(PT=12\).
Mini check

Arc and tangent check

Adjacent arcs are \(128^\circ\) and \(67^\circ\). Two tangents from P have lengths \(3x+1\) and \(5x-11\). Find the combined arc and x.

Show answer and explanation

Answer: \(195^\circ\) and \(x=6\).

Add the arcs; set the tangent lengths equal because they start at the same exterior point.

Key takeaways

Key takeaways

  • A minor arc matches its central-angle measure; a circle totals 360 degrees.
  • Adjacent arcs add and a major arc is 360 degrees minus its corresponding minor arc.
  • A tangent is perpendicular to the radius at contact.
  • Tangents from one exterior point are congruent.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.