Apply central angle, inscribed angle, chord, tangent, and secant relationships in a circle; compute arc length and sector area; and write and use the equation of a circle in the coordinate plane.

What this topic is asking

The Regents Geometry exam (the Circles, G-C, and Expressing Geometric Properties with Equations, G-GPE, clusters) wants you to use the angle relationships in a circle (central, inscribed, and those from chords, tangents, and secants), compute arc length and sector area, and write and use the equation of a circle on the coordinate plane. Circle relationships appear across Part I and the constructed-response parts.

Central and inscribed angles

The two angle types at the heart of circle geometry behave differently.

A central angle (vertex at the center) equals its arc. An inscribed angle (vertex on the circle) is half its intercepted arc. Two consequences are heavily tested: an angle inscribed in a semicircle is a right angle (its arc is 180 degrees, half is 90), and inscribed angles intercepting the same arc are equal.

Chords, tangents, and secants

Further relationships govern segments and the angles they form:

• A tangent line touches the circle once and is perpendicular to the radius at the point of tangency.
• Two chords intersecting inside a circle multiply their segments equally: $a \cdot b = c \cdot d$ for the parts of the two chords.
• A perpendicular from the center to a chord bisects the chord.

These let you solve for unknown segment lengths and angle measures in circle diagrams.

Arc length and sector area

An arc or a sector is a fraction of the whole circle, determined by its central angle.

The equation of a circle

On the coordinate plane, a circle is described by its center and radius.

Read off the center by flipping the sign inside each bracket, and the radius as the square root of the right side. A point lies on the circle exactly when substituting its coordinates makes the equation true. A clarifying point worth stressing is the sign of the center: in $(x + 2)^2 + (y - 5)^2 = 9$, the center is $(-2, 5)$, not $(2, -5)$, because the form subtracts $h$ and $k$. When a circle is given in expanded form, completing the square on the $x$ and $y$ terms recovers this standard form so you can read the center and radius.

Try this

Q1. A central angle measures 72 degrees. What is its intercepted arc? [1 credit]

• Cue. A central angle equals its arc: 72 degrees.

Q2. State the center and radius of $(x - 1)^2 + (y + 4)^2 = 36$. [2 credits]

• Cue. Center $(1, -4)$, radius $\sqrt{36} = 6$.