SAT Math — circle theorems: inscribed angles, tangents and chords

The digital SAT groups its circle questions under the Geometry and Trigonometry domain, and the College Board's skill description asks students to use definitions, properties and theorems about radii, diameters, tangents, angles, arc lengths and sector areas to solve problems. Formulas for area and circumference sit on the reference sheet; the theorems do not. This material trains exactly those theorems, the part of circle work that has to be known rather than looked up.

Each of the twelve quiz questions comes with its own drawing, the way circle questions usually appear on the test, and every question also states its measures in words, so nothing depends on reading values off the picture. The set covers the relationships that come up again and again: an inscribed angle is half the central angle that opens onto the same arc; two inscribed angles on the same arc are equal; an angle inscribed in a semicircle is a right angle; a tangent is perpendicular to the radius at the point where it touches; two tangent segments from one outside point are equal; the perpendicular from the center to a chord bisects it; and opposite angles of a quadrilateral inscribed in a circle add up to 180 degrees.

The questions are built at the level of the harder half of the test rather than as one-step recall. Several give angle measures as algebraic expressions, such as a central angle of (5x + 20) degrees against an inscribed angle of (3x + 4) degrees, so the theorem becomes an equation to solve. Others combine two ideas: a tangent with the Pythagorean theorem, two parallel chords on opposite sides of the center, a chord whose arc height is known, or an inscribed angle that has to be turned into an arc length. One question places the vertex on the minor arc, the case that catches students who halve the wrong arc. Every explanation shows the solution and says where each wrong option comes from, which is usually the mistake the question is designed to reveal.

The flashcards state each theorem in one line and add three quick relationships worth memorizing: the angle between two tangents and the central angle they cut off, the obtuse inscribed angle on the minor arc, and the arc length that follows from an inscribed angle.

The material offers a quiz and flashcards. It is practice written by Zestly and is not affiliated with or endorsed by the College Board.

  • Relate inscribed and central angles on the same arc, including when the vertex lies on the minor arc
  • Use the tangent-radius right angle and equal tangent segments to find lengths
  • Find chord lengths and distances from the center with the perpendicular bisector of a chord
  • Apply the supplementary opposite angles of an inscribed quadrilateral
  • Turn a circle theorem into an equation when angles are given as algebraic expressions

Practice material written by Zestly, based on the College Board Assessment Framework for the Digital SAT Suite (version 3.01, August 2024), Geometry and Trigonometry domain, skill Circles.

Sample question

Circle with center O. Points A, B and C lie on the circle, with C on the larger arc from A to B. Radii OA and OB form a central angle marked 128 degrees; segments CA and CB form the inscribed angle ACB, which is marked but not labeled.

Points $A$, $B$ and $C$ lie on a circle with center $O$, and $C$ is on the major arc $AB$. Central angle $AOB$ measures $128^\circ$. What is the measure, in degrees, of inscribed angle $ACB$?

See the answer

$64$

An inscribed angle is half the central angle that opens onto the same arc. Angle $ACB$ and angle $AOB$ both open onto minor arc $AB$, so angle $ACB = \frac{128^\circ}{2} = 64^\circ$. The value $128$ repeats the central angle; $116$ ($180 - 64$) is the angle for a vertex on the minor arc, which is not where $C$ is; $32$ halves the angle twice.

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