Geometry: Circle Basics

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Questions and Answers

What is a characteristic of a major arc?

It is larger than a semicircle. (correct)

It is a diameter of the circle.

It is exactly half of the circle.

It is smaller than a semicircle.

How is a central angle defined?

An angle whose vertex is on the circumference.

An angle whose vertex is at the center of the circle. (correct)

An angle formed by two chords intersecting inside the circle.

An angle whose sides are tangent to the circle.

What does Circle Theorem 2 state regarding angles?

The angle at the center is equal to the angle at the circumference.

Angles in the same segment are supplementary.

The angle at the circumference is three times the angle at the center.

The angle at the center is twice the angle at the circumference. (correct)

What can be concluded about the angle in a semicircle according to Circle Theorem 4?

It is 90 degrees. Signup and view all the answers

What is the defining property of a secant line in relation to a circle?

It intersects the circle at two points. Signup and view all the answers

According to Circle Theorem 5, what does the perpendicular from the center of a circle to a chord create?

Two equal segments of the chord. Signup and view all the answers

What is the relationship between tangents that meet at the same point on a circle?

They are equal in length. Signup and view all the answers

Which of the following statements about cyclic quadrilaterals is true?

The opposite angles total 180 degrees. Signup and view all the answers

What term describes a line segment from the center to any point on the circle?

Radius Signup and view all the answers

Which formula can be used to calculate the circumference of a circle using the diameter?

C = πd Signup and view all the answers

If the radius of a circle is 5 cm, what is its circumference?

31.4 cm Signup and view all the answers

In what situation would you use S = (θ/360)2πr to calculate arc length?

When θ is in degrees Signup and view all the answers

What is the area of a sector of a circle with a radius of 3 cm and angle of 60 degrees?

4.5 cm² Signup and view all the answers

Which of the following describes a minor arc?

An arc smaller than a semicircle Signup and view all the answers

How does an inscribed angle compare to a central angle that subtends the same arc?

It is half the measure of the central angle Signup and view all the answers

What is the relationship between the radius and diameter of a circle?

Diameter is twice the radius Signup and view all the answers

Study Notes

Circle Definition

A circle is a closed shape formed by all points equidistant from a central point.
Named by its center.

Circle Components

Radius: Line segment from the center to any point on the circle.
Chord: Line segment with endpoints on the circle.
Diameter: A chord passing through the center.

Circumference

Circumference of a circle is the distance around it.
Another name for the perimeter of a circle.
Formulas:
- C = 2πr (where r is the radius)
- C = πd (where d is the diameter)
- π = 3.14 (approximately)

Arc Length

Arc length is the distance along the circumference of a circle or curve (arc).
Formulas:
- S = θr (where θ is the angle in radians)
- S = (θ/360)2πr (where θ is the angle in degrees)

Sector Area

The area enclosed within the boundary of a sector.
Always originates from the center of the circle.
Formulas:
- A = 1/2θr² (where θ is the angle in radians)
- A = (θ/360)(πr²) (where θ is the angle in degrees)

Arcs of a Circle

Semicircle: An arc whose endpoints are the points of the diameter.
Minor Arc: An arc smaller than a semicircle.
Major Arc: An arc larger than a semicircle.

Angles of a Circle

Central Angle: An angle whose vertex is the center of the circle.
Inscribed Angle: An angle whose vertex is on the circle and whose sides contain chords of the circle.

Secant and Tangent

Secant: A line that intersects the circle in two points.
Tangent: A line lying on the same plane that intersects the circle in exactly one point.
Point of Tangency: The point where the tangent touches the circle.

Circle Theorems

Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.
Angles at the Center and Circumference: The angle at the center is twice the angle at the circumference.
Angles in the Same Segment: Angles in the same segment are equal.
Angles in a Semicircle: The angle in a semicircle is 90 degrees.
Chord of a Circle: The perpendicular from the center of a circle to a chord bisects the chord.
Tangent of a Circle:
- The angle between a tangent and radius is 90 degrees.
- Tangents that meet at the same point are equal in length.
Cyclic Quadrilateral: The opposite angles in a cyclic quadrilateral total 180 degrees.

Key Terminology

Center: The central point of the circle.
Radius: The distance from the center to any point on the circle.
Chord: A line segment connecting two points on the circle.
Diameter: The distance across the circle passing through the center.
Circumference: The distance around the circle.
Pi (π): The ratio of the circumference of a circle to its diameter.

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Description

Explore the fundamental concepts of circles, including definitions, components like radius and diameter, and essential formulas for circumference, arc length, and sector area. This quiz will test your understanding of how circles function and are measured in geometry.