Overview

Arcs and chords are closely connected in circle geometry. A central angle determines both the arc it intercepts and the chord connecting the endpoints of that arc.

Many AMC geometry problems can be solved by moving freely between angles, arcs, and chords. In particular, equal chords correspond to equal arcs, and larger arcs correspond to longer chords.

Note: The little arc notation over segment names denotes an arc. For example, with arcs AB and CD,

Definitions and Key Ideas

A chord is a segment whose endpoints lie on a circle.

An arc is a portion of the circle between two points on the circle.

A central angle is an angle whose vertex is at the centre of the circle.

An inscribed angle is an angle whose vertex lies on the circle itself, with both sides as chords.

A tangent to a circle is a line that touches the circle at exactly one point.

If a central angle intercepts an arc of measure $\theta^\circ$, then the arc itself also has measure $\theta^\circ$.

Fundamental Relationships

Equal Chords and Equal Arcs

• Equal chords subtend equal arcs.
• Equal arcs subtend equal chords.
• Equal chords subtend equal central angles.

Thus if $AB = CD$, then $\widehat{AB} = \widehat{CD}$ and the corresponding central angles are equal.

Larger Arcs Correspond to Larger Chords

If one arc has greater measure than another, then its corresponding chord is longer:

Diameter and Semicircles

A diameter divides a circle into two semicircles, each measuring $180^\circ$. Any chord between the endpoints of a diameter is the longest possible chord.

Angle Theorems

Inscribed Angle Theorem

The central angle is exactly twice the inscribed angle when both subtend the same arc:

where $O$ is the centre and $P$ is any point on the major arc. This is one of the most important and frequently tested results in circle geometry.

Corollary. Since all inscribed angles subtending the same arc yield the same central angle, any two inscribed angles subtending the same arc are equal.

Angles in the Same Segment

All inscribed angles subtending the same chord from the same side of that chord are equal:

This follows directly from the Inscribed Angle Theorem, since both angles equal half the same arc.

Angle in a Semicircle (Thales' Theorem)

If $AB$ is a diameter, then any inscribed angle $\angle APB$ with $P$ on the circle satisfies

This is a special case of the Inscribed Angle Theorem: the arc $AB$ is a semicircle of $180°$, so the inscribed angle is $180° / 2 = 90°$. Whenever you see a triangle inscribed in a semicircle, the angle at the circumference is a right angle.

Cyclic Quadrilateral — Opposite Angles

A quadrilateral whose four vertices all lie on a circle is called a cyclic quadrilateral. Its opposite angles are supplementary:

This follows because each pair of opposite angles subtends arcs that together make the full circle of $360°$, so each angle is half of $180°$.

Converse. If the opposite angles of a quadrilateral sum to $180°$, then the quadrilateral is cyclic.

Tangent-Chord Angle (Alternate Segment Theorem)

The angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle subtending the same chord from the opposite arc:

This is also called the tangent-chord angle or alternate segment theorem. It appears frequently in problems where a tangent is drawn at one end of a chord.

Tangent Theorems

Radius to Tangent is Perpendicular

A tangent to a circle is perpendicular to the radius drawn to the point of tangency:

This is the key fact for computing distances involving tangents.

Two Tangents from an External Point

[diagram: tangent_properties_and_intersecting_chords — left panel]

If two tangent segments are drawn to a circle from the same external point $P$, touching the circle at $T_1$ and $T_2$, then they are equal in length:

The triangles $OT_1P$ and $OT_2P$ are congruent (hypotenuse $OP$ and equal radii $OT_1 = OT_2$, with right angles at the touch points), giving the result immediately.

Chord Intersection Theorems

Intersecting Chords Inside the Circle

If two chords $AB$ and $CD$ intersect at a point $P$ inside the circle, then the products of their segments are equal:

This follows from the similar triangles formed by the chords, which arise because the inscribed angles subtending the same arc are equal.

Secant-Secant from an External Point

If two secants are drawn from an external point $P$, meeting the circle at $A$, $B$ and $C$, $D$ respectively (with $A$, $C$ the nearer points), then:

Secant-Tangent from an External Point

If a tangent from external point $P$ touches the circle at $T$, and a secant through $P$ meets the circle at $A$ and $B$ (with $A$ nearer), then:

This can be seen as the limiting case of the secant-secant theorem when the two intersection points of one secant merge into a single tangent point.

Chords and Symmetry

If a radius is drawn perpendicular to a chord, then it bisects the chord. Conversely, if a radius bisects a chord, it is perpendicular to the chord:

A radius perpendicular to a chord bisects the chord.

Arc Length

Arc measure is in degrees; arc length is in units of distance. For a circle of radius $r$ and central angle $\theta$ in radians:

Always convert degrees to radians before applying this formula. For example, $60° = \pi/3$.

Worked Examples

Example: Chord from a 60° angle

A circle has radius $10$ and central angle $60°$. Find the chord length.

The triangle formed by the two radii and the chord has two sides of length $10$ with an included angle of $60°$. By the isoceles triangle, all three angles equal $60°$, so the triangle is equilateral and the chord length is $\boxed{10}$.

Example: Arc Length

Find the arc length when $r = 6$ and $\theta = 150°$.

Convert: $150° = 5\pi/6$. Then $s = 6 \cdot \tfrac{5\pi}{6} = 5\pi$.

Example: Using the Inscribed Angle Theorem

A central angle $\angle AOB = 100°$. Find the inscribed angle $\angle APB$ for a point $P$ on the major arc.

Example: Cyclic Quadrilateral

In a cyclic quadrilateral $ABCD$, $\angle A = 110°$. Find $\angle C$.

Example: Two Tangents

From external point $P$, two tangents touch a circle of radius $5$ at $T_1$ and $T_2$. If $PT_1 = 12$, find $PO$ (distance to centre).

By the radius-tangent perpendicularity, $OT_1 \perp PT_1$, so:

Example: Intersecting Chords

Two chords intersect inside a circle. One chord is divided into segments of length $4$ and $6$. One segment of the other chord has length $3$. Find the other segment.

Example: Comparing Chords

Chord $AB$ subtends $120°$ and chord $CD$ subtends $80°$. Then $AB > CD$ since a larger arc corresponds to a longer chord.

Example: Perpendicular Radius

A radius bisects a chord of length $12$. Each half has length $6$.

Example: Semicircle Arc Length

A circle has radius $8$. The semicircular arc length is $s = 8\pi$.

Strategy Checklist

• Identify whether angles are central or inscribed, then apply the factor-of-two relationship.
• Check for cyclic quadrilaterals — opposite angles sum to $180°$.
• Look for tangent points — the radius to the tangent is always perpendicular.
• For two tangents from an external point, use equal tangent lengths.
• When chords intersect, use the product-of-segments formula.
• Ask whether a right angle appears — if so, look for a diameter (Thales' theorem).
• Determine whether the problem asks for arc measure or arc length; convert degrees to radians for the latter.
• Look for equal chords and equal arcs.
• Search for symmetry involving radii and chords.

Common Pitfalls

• Forgetting the factor of 2 between central and inscribed angles.
• Applying the inscribed angle theorem when the centre and inscribed point are on the same arc (the formula changes sign — always check which arc the point lies on).
• Confusing arc measure with arc length, or using degrees directly in $s = r\theta$.
• Assuming equal arcs in different circles imply equal chord lengths.
• Forgetting that a diameter is the longest chord.
• Missing that a radius perpendicular to a chord bisects it.
• Forgetting the radius-to-tangent perpendicularity when setting up right triangles.

Practice Problems