Geometry Chapter 9 Flashcards

Circle Concepts

• A circle consists of all points in a plane at a fixed distance from a central point.
• The center is the specific point located in the middle of the circle.
• A radius is the segment connecting the center to any point on the circle, representing the circle's distance from the center.
• The diameter is a special chord that passes through the center, effectively the longest distance across the circle.
• A chord connects two points on the circle without passing through the center.

Lines Related to Circles

• A secant is a line that intersects a circle in two points and contains a chord.
• A tangent touches the circle at exactly one point, known as the point of tangency.

Special Circle Relations

• Congruent circles have equal radii.
• Concentric circles share the same center but differ in radius.
• Inscribed polygons fit within circles, while circumscribed circles surround polygons.

Theorems and Properties

• Theorem 9-1 states that a tangent is perpendicular to the radius at the point of tangency.
• Corollary to the theorem indicates that tangents from a single external point are congruent.
• Theorem 9-2 explains that a line perpendicular to a radius at its endpoint is a tangent.

Tangents and Circle Interactions

• Common interior tangents run between two circles, touching them on opposite sides.
• Common exterior tangents touch both circles on the same side.
• Externally tangent circles touch at one point on shared sides, whereas internally tangent circles have one circle within the other touching at a single point.

Arcs and Angles

• A central angle is defined by an angle at the circle's center, while an arc represents part of the circle's circumference.
• The measure of an arc is determined by its central angle; minor arcs are under 180°, and major arcs range between 180° and 360°.

Arc Relationships

• Adjacent arcs share precisely one common point.
• The Arc Addition Postulate states that the sum of the measures of two adjacent arcs equals the measure of the arc they form together.
• Congruent arcs have equal measures within the same circle or within congruent circles.

Chord Theorems

• Theorem 9-4 establishes that congruent arcs correspond to congruent chords and vice versa.
• Theorem 9-5 states a diameter that is perpendicular to a chord splits the chord and its corresponding arc in half.
• Theorem 9-6 states that chords at equal distances from the center are congruent.

Inscribed Angles

• An inscribed angle has its vertex on the circle and its sides as chords; it measures half of its intercepted arc.
• If two inscribed angles intercept the same arc, they are congruent.
• An inscribed angle in a semicircle is always a right angle.

Angle Measurements

• The angle formed by a chord and a tangent equals half the measure of the intercepted arc.
• For angles formed by intersecting chords inside a circle, the measurement is half the sum of the intercepted arcs.
• Theorem 9-10 provides angle relationships for secants and tangents, often yielding expressions for angles dependent on arc lengths.

Secants and Tangents

• Relationships among secants and tangents can be represented by various equations, commonly involving segment lengths derived from these intersecting lines.
• The Corollary Theorem 9.1 specifies that tangents from a common external point are equal in length.

Summary of Product Relationships

• Theorem 9-11 indicates that the products of the lengths of segments formed by intersecting chords are equal.
• Theorem 9-12 presents a similar relationship for secants.
• Theorem 9-13 relates the length of a secant and a tangent to lengths formed by parts of these lines.