🔹 Center

The fixed point at the middle of the circle, equidistant from all points on the circle.

🔹 Radius (r)

A line segment from the center to any point on the circle.

🔹 Diameter (d)

A line segment passing through the center with endpoints on the circle. It is the longest chord.

d = 2r

🔹 Chord

A line segment with both endpoints on the circle.

🔹 Arc

A curved section of the circle's circumference between two points.

• Minor Arc: The shorter arc between two points (less than 180°)

• Major Arc: The longer arc between two points (greater than 180°)

🔹 Secant

A line that intersects the circle at two points.

🔹 Tangent

A line that touches the circle at exactly one point (point of tangency).

🔹 Sector

A region bounded by two radii and an arc.

🔹 Segment

A region bounded by a chord and an arc.

What is a Central Angle?

A central angle is an angle whose vertex is at the center of the circle and whose sides are radii.

🔹 Key Property

The measure of a central angle = The measure of its intercepted arc

If central angle = 60°, then the intercepted arc = 60°

What is Arc Length?

Arc length is the distance along the curved line of an arc.

🔹 Formula (Degrees)

Arc Length = (θ/360°) × 2πr

θ = Central angle in degrees

r = Radius of the circle

🔹 Formula (Radians)

Arc Length (s) = rθ

θ = Central angle in radians

r = Radius of the circle

Key Relationship

π radians = 180°

🔹 Degrees to Radians

Radians = Degrees × (π/180°)

🔹 Radians to Degrees

Degrees = Radians × (180°/π)

What is a Sector?

A sector is a pie-shaped region bounded by two radii and an arc.

🔹 Formula (Degrees)

Area of Sector = (θ/360°) × πr²

θ = Central angle in degrees

r = Radius of the circle

🔹 Formula (Radians)

Area of Sector = (1/2)r²θ

θ = Central angle in radians

r = Radius of the circle

🔹 Theorem 1

In the same circle or congruent circles, congruent chords have congruent arcs.

🔹 Theorem 2

In the same circle or congruent circles, congruent arcs have congruent chords.

🔹 Theorem 3

A perpendicular from the center of a circle to a chord bisects the chord.

🔹 Tangent-Radius Theorem

A tangent line is PERPENDICULAR to the radius at the point of tangency

(Forms a 90° angle)

🔹 Two Tangent Theorem

If two tangent segments are drawn to a circle from an external point, then:

• The tangent segments are CONGRUENT (equal length)

• The angle formed by the two tangents is bisected by the line joining the external point to the center

📝 Example

If tangent segments PA and PB are drawn from point P to circle O, then PA = PB

What is an Inscribed Angle?

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle.

🔹 Inscribed Angle Theorem

Inscribed Angle = (1/2) × Central Angle

OR

Inscribed Angle = (1/2) × Intercepted Arc

🔹 Corollary: Inscribed Angle in Semicircle

An angle inscribed in a semicircle is a RIGHT ANGLE (90°)

This happens when the inscribed angle intercepts a diameter

🔹 Inscribed Angles on Same Arc

All inscribed angles that intercept the same arc are CONGRUENT (have equal measures).

What is an Inscribed Quadrilateral?

An inscribed quadrilateral (cyclic quadrilateral) is a quadrilateral with all four vertices on a circle.

🔹 Opposite Angles Theorem

Opposite angles of an inscribed quadrilateral are SUPPLEMENTARY

∠A + ∠C = 180°

∠B + ∠D = 180°

🔹 Two Chords Intersecting INSIDE the Circle

Angle = (1/2)(Arc 1 + Arc 2)

The angle equals half the sum of the two intercepted arcs.

🔹 Two Secants Intersecting OUTSIDE the Circle

Angle = (1/2)|Arc 1 - Arc 2|

The angle equals half the difference of the two intercepted arcs.

🔹 Secant and Tangent Intersecting OUTSIDE

Angle = (1/2)|Arc 1 - Arc 2|

The angle equals half the difference of the two intercepted arcs.

🔹 Tangent and Chord at Point of Tangency

Angle = (1/2) × Intercepted Arc

The angle between a tangent and a chord equals half the intercepted arc.

Power of a Point Theorems

These theorems relate the lengths of segments formed when lines intersect circles.

🔹 Two Chords Intersecting INSIDE the Circle

a × b = c × d

If two chords intersect at point P, then (segment 1 of chord 1) × (segment 2 of chord 1) = (segment 1 of chord 2) × (segment 2 of chord 2)

🔹 Two Secants from External Point

(whole × external) = (whole × external)

PA × PB = PC × PD

(Whole secant) × (External part) = (Whole secant) × (External part)

🔹 Secant and Tangent from External Point

(Tangent)² = (whole secant) × (external part)

PT² = PA × PB

The square of the tangent segment equals the product of the secant and its external part.

🔹 Basic Measurements

🔹 Angle Relationships