Podcast
Questions and Answers
What is a circle?
What is a circle?
A collection of points that are equidistant from a center point.
What does 'NOT in the same plane' refer to?
What does 'NOT in the same plane' refer to?
- Sphere (correct)
- Point
- Line
- Circle
What does 'IN the same plane' refer to?
What does 'IN the same plane' refer to?
- Point
- Circle (correct)
- Line
- Sphere
What is a sphere?
What is a sphere?
What is a radius?
What is a radius?
What is a chord?
What is a chord?
What is a diameter?
What is a diameter?
What is a secant?
What is a secant?
What is a tangent?
What is a tangent?
What are congruent circles?
What are congruent circles?
What is a central angle?
What is a central angle?
What are minor arcs?
What are minor arcs?
What is a semi-circle?
What is a semi-circle?
What are major arcs?
What are major arcs?
What is the arc addition postulate?
What is the arc addition postulate?
What is the formula for arc length?
What is the formula for arc length?
What are concentric circles?
What are concentric circles?
What does Theorem 12-1 state?
What does Theorem 12-1 state?
What is the Converse of Theorem 12-1?
What is the Converse of Theorem 12-1?
What does Theorem 12-2 state?
What does Theorem 12-2 state?
What does Theorem 12-5 state?
What does Theorem 12-5 state?
What does Theorem 12-7 state?
What does Theorem 12-7 state?
What does Theorem 12-8 state?
What does Theorem 12-8 state?
What does Theorem 12-9 state?
What does Theorem 12-9 state?
What does Theorem 12-10 state?
What does Theorem 12-10 state?
What is an inscribed angle?
What is an inscribed angle?
What does Corollary 1 state?
What does Corollary 1 state?
What does Corollary 2 state?
What does Corollary 2 state?
What does Corollary 3 state?
What does Corollary 3 state?
What is the interior angle formula?
What is the interior angle formula?
What is the exterior angle formula?
What is the exterior angle formula?
What is the standard form of a circle?
What is the standard form of a circle?
What are some examples of Pythagorean triples?
What are some examples of Pythagorean triples?
Study Notes
Circle Concepts
- A circle is defined as a collection of points equidistant from a central point.
- A sphere consists of points in space that are equidistant from a central point but not in the same plane as a circle.
Circle Elements
- Radius: The segment connecting the center of the circle to a point on its circumference.
- Chord: A segment that joins two points on a circle.
- Diameter: A special type of chord that passes through the center of the circle.
- Secant: A line that intersects a circle and contains a chord.
- Tangent: A line that touches a circle at exactly one point.
Arc Definitions
- Central angle: Formed by two radii at the center; its measure is equal to the intercepted arc.
- Minor arcs: Measures between 0° and 180°.
- Semi circle: An arc that measures exactly 180°.
- Major arcs: Measures between 180° and 360°.
Arc and Length Formulas
- Arc addition postulate: The measure of an arc formed by two adjacent radii is the sum of the measures of the two arcs they intercept.
- Arc length formula: Calculated using the formula ((x/360) \times 2\pi r).
Special Circle Relations
- Concentric circles: Multiple circles sharing the same center.
- Theorem 12-1: If a line is perpendicular to a circle, it intersects the radius at a right angle at the point of tangency.
- Converse of Theorem 12-1: A line that is perpendicular to a radius at its endpoint is a tangent.
Circle Theorems
- Theorem 12-2: If two tangents are drawn from a common exterior point, the line segment connecting the point to the tangency points forms equal lengths.
- Theorem 12-5: In congruent circles, congruent circles correspond to congruent arcs and congruent chords.
- Theorem 12-7: Congruent chords are equidistant from the center in both congruent and same circles.
- Theorem 12-8: A diameter that is perpendicular to a chord bisects both the chord and the arc it intercepts.
- Theorem 12-9: A diameter that bisects a chord is perpendicular to it.
- Theorem 12-10: The perpendicular bisector of a chord passes through the center of the circle.
Inscribed Angles and their Properties
- Inscribed angle: Equal to half the measure of its intercepted arc.
- Corollary 1: Inscribed angles sharing a common arc are congruent.
- Corollary 2: An inscribed angle in a semicircle is always 90°.
- Corollary 3: Opposite angles of an inscribed quadrilateral are supplementary.
Angle Formulas
- Interior angle formula: (\frac{1}{2} \times (\text{sum of intercept arc + arc})).
- Exterior angle formula: (\frac{1}{2} \times (\text{large arc} - \text{small arc})).
Circle Equation
- The standard form of a circle is given by the equation ((x-h)^2 + (y-k)^2 = r^2), where ((h, k)) is the center and (r) is the radius.
Pythagorean Triples
- Known Pythagorean triples include (5, 12, 13), (3, 4, 5), and (7, 24, 25).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Prepare for your Geometry Chapter 12 test with these flashcards. Each card features key terms and definitions related to circles and spheres, helping you grasp the essential concepts. Use these flashcards to reinforce your understanding and excel in your exam.
