Questions and Answers
What is a circle?
What is the center of a circle?
The given point in the middle of the circle.
What is the definition of a radius?
What is a diameter?
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What is a chord?
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What is a secant?
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What is a common chord?
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What is a tangent in relation to a circle?
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What is an arc?
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The measure of a minor arc is less than ____.
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What is a central angle?
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What does Theorem 9-1 state?
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What is a congruent circle?
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Match the following terms with their definitions:
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What does the Arc Addition Postulate state?
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What does Theorem 9-11 state regarding two chords?
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Study Notes
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.
Studying That Suits You
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Description
Test your knowledge on key geometry concepts with these flashcards from Chapter 9. Each flashcard covers important terminology related to circles, including definitions of terms like center, radius, and diameter. Perfect for studying and reinforcing your understanding of geometric principles.
