Ücretsiz Kullanmaya Başla

Arcs, chords, tangents, inscribed angles Knowledge-Based Study

Math · Geometry · Math · Circles

Math topic explanations on Arcs, chords, tangents, inscribed angles.

Study notes

  1. Basic Circle Elements: Arcs and Chords

    A chord is a line segment whose endpoints both lie on a circle. A secant is a line that intersects a circle at two distinct points, extending indefinitely in both directions. An arc is an unbroken part of the circle's circumference.

    • Minor Arc: An arc whose measure is strictly less than 180°. It is typically named using its two endpoints, such as AB^\widehat{AB}.
    • Major Arc: An arc whose measure is strictly greater than 180°. It is named using three points—the endpoints and an intermediate point—such as ACB^\widehat{ACB}.
    • Semicircle: An arc whose endpoints are the endpoints of a diameter, with a measure of exactly 180°.

    Fundamental Chord and Arc Theorems

    Within the same circle or in congruent circles, the following geometric properties hold:

    • Congruent Chords and Arcs: Two chords are congruent if and only if their corresponding minor arcs have equal measures: AB=CD  ⟺  m(AB^)=m(CD^)AB = CD \iff m(\widehat{AB}) = m(\widehat{CD})
    • Perpendicular Bisector of a Chord: A radius or diameter that is perpendicular to a chord bisects that chord and its intercepted arc. Conversely, the perpendicular bisector of any chord passes through the center of the circle.
    • Equidistant Chords: Two chords are congruent if and only if they are equidistant from the circle's center.

    Angle Measures in Circles

    An angle related to a circle can have its vertex at the center, on the circumference, inside the circle, or outside the circle.

    Central Angle

    A central angle has its vertex at the center of the circle. The degree measure of a central angle is equal to the degree measure of its intercepted arc: m(∠AOB)=m(AB^)m(\angle AOB) = m(\widehat{AB})

    Inscribed Angle Theorem

    An inscribed angle has its vertex on the circle, and its sides contain chords of the circle. The measure of an inscribed angle is half the measure of its intercepted arc: m(∠APB)=12m(AB^)m(\angle APB) = \frac{1}{2}m(\widehat{AB})

    Key Corollaries

    • Inscribed Angles Subtending the Same Arc: Inscribed angles that intercept the same arc (or congruent arcs) are congruent. If ∠APB\angle APB and ∠AQB\angle AQB both intercept AB^\widehat{AB}, then: m(∠APB)=m(∠AQB)m(\angle APB) = m(\angle AQB)
    • Thales's Theorem (Angle in a Semicircle): An inscribed angle that intercepts a diameter or semicircle is always a right angle (90°): m(∠APB)=90∘when AB is a diameterm(\angle APB) = 90^\circ \quad \text{when } AB \text{ is a diameter}
    • Cyclic Quadrilaterals: In any quadrilateral inscribed in a circle, opposite angles are supplementary (their sum is 180°).

    Tangents and Their Properties

    A tangent is a line in the plane of a circle that intersects the circle at exactly one point, known as the point of tangency.

    Radius-Tangent Perpendicularity

    A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency: r⊥tat the point of contact Tr \perp t \quad \text{at the point of contact } T

    Tangent Segments from an External Point

    When two tangent segments are drawn to a circle from a common external point PP, tangent to points AA and BB on the circle:

    • The tangent segments are equal in length: PA=PBPA = PB
    • The line segment connecting the external point to the center OO bisects the angle formed by the two tangents: m(∠APO)=m(∠BPO)m(\angle APO) = m(\angle BPO).
    • The central quadrilateral PAOBPAOB formed by the radii and tangent segments has two right angles, meaning the angle between the tangents and the central angle are supplementary: m(∠APB)+m(∠AOB)=180∘m(\angle APB) + m(\angle AOB) = 180^\circ

    Angles Formed by Intersecting Lines

    The position of the vertex determines how the angle's measure relates to its intercepted arcs.

    1. Tangent-Chord Angle (Vertex on the Circle)

    An angle formed by a tangent and a chord meeting at the point of tangency measures half the intercepted arc: m(∠TAB)=12m(AB^)m(\angle TAB) = \frac{1}{2}m(\widehat{AB})

    2. Interior Intersection (Chords Intersecting Inside the Circle)

    When two chords intersect inside a circle, the measure of each vertically opposite angle is half the sum of the measures of the intercepted arcs: m(∠1)=m(AB^)+m(CD^)2m(\angle 1) = \frac{m(\widehat{AB}) + m(\widehat{CD})}{2}

    3. Exterior Intersection (Secants and Tangents Intersecting Outside the Circle)

    When two lines (secants or tangents) intersect outside the circle, the measure of the angle formed is half the difference of the intercepted arcs: m(∠P)=m(far^)−m(near^)2m(\angle P) = \frac{m(\widehat{\text{far}}) - m(\widehat{\text{near}})}{2}

    • Two Secants: m(∠P)=12(m(AC^)−m(BD^))m(\angle P) = \frac{1}{2}\left(m(\widehat{AC}) - m(\widehat{BD})\right)
    • Secant and Tangent: m(∠P)=12(m(AC^)−m(BC^))m(\angle P) = \frac{1}{2}\left(m(\widehat{AC}) - m(\widehat{BC})\right)
    • Two Tangents: m(∠P)=12(m(ACB^)−m(AB^))=180∘−m(AB^)m(\angle P) = \frac{1}{2}\left(m(\widehat{ACB}) - m(\widehat{AB})\right) = 180^\circ - m(\widehat{AB})