Chapter 3 Review — Part 2: The Circle

This second part is all about the circle: its parts, its angles and arcs, chords, tangents and secants, and finally its equation. Each block below is a compact recap with its main rules.

The circle Parts of a circle

The radius $r$ reaches from the center to the circle; the diameter $d$ crosses through the center; a chord joins two points without passing through it.

Circumference $= 2\pi r = \pi d$ . The diameter is the longest chord.

Central angles Central angles and arc length

A central angle has its vertex at the center. Non-overlapping central angles add to 360°, and each one cuts off a minor arc and a major arc.

The arc is the same fraction of the circumference as the angle is of 360°:

\text{arc length} = \frac{\theta}{360^\circ}\times 2\pi r

With

r = 5

and

\theta = 90^\circ

: arc

= \tfrac{90}{360}\times 2\pi\times 5

⟹ arc ≈ 7.85.

Inscribed angles Angles on the circle

An inscribed angle has its vertex on the circle, and it equals half its intercepted arc — so the central angle is twice the inscribed angle on the same arc.

Two inscribed angles on the same arc are equal.

An inscribed angle on a diameter (a semicircle) is 90°.

In a cyclic quadrilateral, opposite angles are supplementary (add to 180°).

Chords Chords and the perpendicular diameter

Congruent chords cut off congruent minor arcs.

A diameter perpendicular to a chord bisects it.

The perpendicular bisector of any chord passes through the center.

When two chords cross inside the circle, the angle is half the sum of the two arcs, and the products of the pieces are equal: $a \times b = c \times d$ .

Tangents Tangents and secants

A tangent meets the circle at one point and is perpendicular to the radius there.

Two tangents from an external point are equal in length.

A tangent–chord angle equals half its intercepted arc.

Vertex outside (two secants, tangent & secant, or two tangents): angle = half the difference of the arcs.

Lengths from an external point

Two secants: whole × external is equal for both —

s_1 \times e_1 = s_2 \times e_2

Tangent and secant: the tangent squared equals whole × external —

t^{2} = s \times e

Equation The equation of a circle

Center at the origin: $x^{2} + y^{2} = r^{2}$ . Center at $(a, b)$ : $(x-a)^{2} + (y-b)^{2} = r^{2}$ . The center's signs are reversed in the equation, and the right side is $r^{2}$ , the radius squared — not the diameter.

Example 1 — center (2, 3), radius 5

Both coordinates are positive, so both signs flip to minus.

⟹ (x − 2)² + (y − 3)² = 25.

Example 2 — center (−2, −3), radius 5

Both coordinates are negative, so both signs flip to plus.

⟹ (x + 2)² + (y + 3)² = 25.

Reference Circle theorems at a glance

Theorem	Rule
Arc length	(θ ÷ 360) × 2πr
Inscribed angle	½ × its intercepted arc
Angle on a diameter	90°
Cyclic quadrilateral	opposite angles sum to 180°
Two chords inside	a × b = c × d
Two secants outside	whole × external = whole × external
Tangent & secant outside	tangent² = whole × external
Equation of a circle	(x − a)² + (y − b)² = r²

Summary

A central angle has its vertex at the center; an inscribed angle sits on the circle and is half the central one on the same arc.
An angle on a diameter is 90°, and a cyclic quadrilateral's opposite angles add to 180°.
Vertex inside a circle: angle = half the sum of the arcs; vertex outside: half the difference.
Lengths: inside, multiply the two pieces; outside, multiply whole by external (and the tangent is squared).
Equation of a circle: reverse the center's signs, and the right side is the radius squared.