Geometry Chapter 12 Flashcards

Questions and Answers

What is a chord?

A segment whose endpoints are on the circle.

What is a secant?

A line that intersects a circle at two points.

What is a tangent?

A line that intersects a circle at exactly one point.

What are congruent circles?

Congruent radii. Signup and view all the answers

What are concentric circles?

Coplanar circles with the same center. Signup and view all the answers

What are tangent circles?

Coplanar circles that intersect at exactly one point. Signup and view all the answers

What is a common tangent?

A line that touches two circles, either crossing in between (internal) or lying outside (external). Signup and view all the answers

What does Theorem 12-1 state?

If a line is tangent to a circle, then it is perpendicular to the radius of the circle. Signup and view all the answers

What does Theorem 2 (12-1) state?

If two segments are tangent to a circle from the same external point, then the segments are congruent. Signup and view all the answers

What is a central angle?

An angle whose vertex is the center of the circle. Signup and view all the answers

What defines minor and major arcs?

Minor arc: an arc whose points are on the interior of the central angle. Major arc: an arc whose points are on the exterior of the central angle. Signup and view all the answers

What does Theorem (12-2) cover?

In a circle or congruent circles: 1) congruent central angles have congruent chords, 2) congruent chords have congruent arcs, 3) congruent arcs have congruent central angles. Signup and view all the answers

What does Theorem 2 (12-2) state?

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

What is a sector of a circle?

The region bounded by two radii of the circle and their intercepted arc. Signup and view all the answers

What is a segment of a circle?

The region bounded by an arc and its chord. Signup and view all the answers

How do you calculate the length of an arc?

2π x radius x measure of the angle / 360. Signup and view all the answers

What are inscribed angles?

An angle whose vertex is on a circle and whose sides contain chords of the circle. Signup and view all the answers

What does the inscribed angles theorem state?

The measure of an inscribed angle is 1/2 the measure of its intercepted arc. Signup and view all the answers

What does Corollary 1 (12-4) state?

If all inscribed angles are connected to the same endpoints of the angle, then they are congruent. Signup and view all the answers

What does Theorem 2 (12-4) state?

An inscribed angle subtends a semicircle if and only if the angle is a right angle. Signup and view all the answers

What does Theorem 3 (12-4) cover?

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Signup and view all the answers

What does Theorem 1 (12-5) state?

The measure of an angle is 1/2 the measure of the arc. Signup and view all the answers

What does Theorem 2 (12-5) state?

The measure of an angle formed inside is 1/2 the sum of the measures of the arcs. Signup and view all the answers

What does Theorem 3 (12-5) state?

The measure of an angle formed outside is 1/2 the difference of the arcs. Signup and view all the answers

What does the chord chord product theorem state?

If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. Signup and view all the answers

What does the secant secant product theorem state?

Whole x outside = whole x outside. Signup and view all the answers

What does the secant tangents products theorem state?

Whole x outside = (tangent) squared. Signup and view all the answers

Study Notes

Key Definitions and Concepts

Chord: A line segment with both endpoints on the circle.
Secant: A line intersecting the circle at two distinct points.
Tangent: A line that touches the circle at a single point, known as the point of tangency.

Circle Relationships

Congruent Circles: Circles with equal radius measurements.
Concentric Circles: Circles that share the same center but have different radii.
Tangent Circles: Two circles that intersect at exactly one point.
Common Tangent: A line tangent to two circles; can be internal (between circles) or external (outside both circles).

Fundamental Theorems

Theorem 12-1: A tangent to a circle is always perpendicular to the radius at the point of tangency.
Theorem 2 (12-1): Two tangent segments drawn from a single external point to a circle are congruent.
Theorem 12-2: In a circle, congruent central angles yield congruent chords, which in turn yield congruent arcs.
Theorem 2 (12-2): A radius or diameter that is perpendicular to a chord bisects the chord and its intercepted arc.

Circle Measurements

Central Angle: An angle located at the center of the circle.
Minor Arc: An arc formed by points within a central angle.
Major Arc: An arc formed by points outside a central angle.
Sector of a Circle: The area between two radii and their intercepted arc; calculated as (π × radius² × angle measure / 360).
Segment of a Circle: The area between a chord and its intercepted arc; determined by subtracting the area of the triangle from the area of the sector.

Arc Length Calculation

Length of an Arc: Computed using the formula (2π × radius × angle measure / 360).

Inscribed Angles and Theorems

Inscribed Angles: Angles with vertices on the circle and sides that are chords.
Inscribed Angles Theorem: The angle measure is half that of its intercepted arc.
Corollary 1 (12-4): All inscribed angles with the same endpoints are congruent.
Theorem 2 (12-4): An inscribed angle subtends a semicircle if, and only if, it measures 90 degrees.
Theorem 3 (12-4): In an inscribed quadrilateral, opposite angles are supplementary.

Additional Angle Theorems

Theorem 1 (12-5): An angle formed outside a circle is equal to one-half the measure of its intercepted arc.
Theorem 2 (12-5): The measure of an angle inside a circle is one-half the sum of the arcs it intercepts.
Theorem 3 (12-5): An angle formed outside the circle is one-half the difference of the measures of the intercepted arcs.

Chords and Secants Theorems

Chord-Chord Product Theorem: The products of the lengths of the segments created by two intersecting chords within a circle are equal.
Secant-Secant Product Theorem: The product of the whole length and the external part of one secant equals the product of the whole and external part of another secant.
Secant-Tangent Product Theorem: The product of the secant's whole length and its external length equals the square of the tangent length.

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Description

Test your knowledge of key terms from Geometry Chapter 12 with these flashcards. Learn the definitions of important concepts like chords, secants, and tangents as well as the properties of congruent and concentric circles. Perfect for students looking to reinforce their understanding of circle geometry.