Geometry Chapter 12 Test Flashcards

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Questions and Answers

What is a circle?

A collection of points that are equidistant from a center point.

What does 'NOT in the same plane' refer to?

Sphere (correct)

Point

Line

Circle

What does 'IN the same plane' refer to?

Point

Circle (correct)

Line

Sphere

What is a sphere?

A set of points in space equidistant from a central point. Signup and view all the answers

What is a radius?

A segment from the center to a point on the circle. Signup and view all the answers

What is a chord?

A segment that joins two points on a circle. Signup and view all the answers

What is a diameter?

A chord that passes through the center. Signup and view all the answers

What is a secant?

A line that contains a chord. Signup and view all the answers

What is a tangent?

A line that intersects a circle in a single point. Signup and view all the answers

What are congruent circles?

Two or more circles with the same radii. Signup and view all the answers

What is a central angle?

An angle formed by the two radii with the vertex at the center. Signup and view all the answers

What are minor arcs?

Arcs between 0 and 180 degrees. Signup and view all the answers

What is a semi-circle?

180 degrees. Signup and view all the answers

What are major arcs?

Arcs between 180 and 360 degrees. Signup and view all the answers

What is the arc addition postulate?

The measure of an arc formed by two adjacent radii is equal to the sum of the two arcs. Signup and view all the answers

What is the formula for arc length?

x/360 times 2(pi)r Signup and view all the answers

What are concentric circles?

Two or more circles with the same center. Signup and view all the answers

What does Theorem 12-1 state?

If a line is perpendicular to a circle, then the radius is perpendicular at the point of tangency. Signup and view all the answers

What is the Converse of Theorem 12-1?

If a line is perpendicular to a radius at its endpoint, then it is a tangent. Signup and view all the answers

What does Theorem 12-2 state?

If two lines are tangent to a circle from a common exterior point, then the segment joins the point to the point of tangency. Signup and view all the answers

What does Theorem 12-5 state?

In the same or congruent circles, congruent circles equal congruent arcs; congruent arcs equal congruent chords. Signup and view all the answers

What does Theorem 12-7 state?

In the same or congruent circles, congruent chords are equidistant from the center. Chords equidistant from the center are congruent. Signup and view all the answers

What does Theorem 12-8 state?

In a circle, if the diameter is perpendicular to a chord, then the diameter bisects the chord and the diameter bisects the arc. Signup and view all the answers

What does Theorem 12-9 state?

If a diameter bisects a chord, then it is perpendicular to the chord. Signup and view all the answers

What does Theorem 12-10 state?

The perpendicular bisector of the chord contains the center. Signup and view all the answers

What is an inscribed angle?

1/2 the arc. Signup and view all the answers

What does Corollary 1 state?

Inscribed angles in a circle that share a common arc are congruent. Signup and view all the answers

What does Corollary 2 state?

An inscribed angle in a semi-circle must be 90 degrees. Signup and view all the answers

What does Corollary 3 state?

Opposite angles of an inscribed quadrilateral are supplementary. Signup and view all the answers

What is the interior angle formula?

1/2 (sum of intercept + arc) Signup and view all the answers

What is the exterior angle formula?

1/2 (large arc - small arc) Signup and view all the answers

What is the standard form of a circle?

(x-h)^2 + (y-k)^2 = r^2 Signup and view all the answers

What are some examples of Pythagorean triples?

5,12,13; 3,4,5; 7,24,25. Signup and view all the answers

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).

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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.