The document defines parts of a circle and properties of circles. It discusses central angles, arcs, chords, and circumferential angles. Several examples are provided to illustrate properties such as if two arcs are equal, their central angles and chords are also equal. Theorems are stated about central angles corresponding to arcs and circumferential angles corresponding to arcs being equal to half the central angle.
When light passes from one medium to another, such as from air to glass, it changes direction. This bending of light is called refraction. Refraction occurs because light slows down when moving into a denser medium like glass or water. The denser the second medium, the greater the bending of light. If light travels from a less dense to a more dense medium, it bends toward the normal line. If it travels from more dense to less dense, it bends away from the normal line. This bending allows for phenomena like mirages and the functioning of lenses and prisms.
This document discusses the phenomenon of refraction of light. It begins by explaining that light travels in a straight line through air but can be reflected or refracted at the interface between two substances. When light passes from one medium to another of different density, it changes speed and bends. This bending is called refraction. The direction of bending depends on whether the light is entering a less or more dense medium. Refraction allows lenses and prisms to bend light in useful ways. The document concludes with descriptions of two hands-on activities to demonstrate refraction.
This document outlines three physics experiments on light reflection. Experiment 1 examines diffuse and regular (specular) reflection using a mirror and matte paper. Experiment 2 demonstrates the laws of reflection using a mirror and graduated disk to measure angles of incidence and reflection. It shows that the angle of incidence equals the angle of reflection, and that the incident, reflected, and normal rays lie in the same plane. Experiment 3 reverses the path of light to show the reversibility of reflection - the path of light does not change if the direction of travel is reversed.
This document provides an overview of key concepts in geometry related to circles such as identifying segments and lines that intersect circles, using properties of tangents to solve problems, finding measures of arcs and angles, writing equations of circles, and using coordinate proofs involving circles. Examples and practice problems are included to help teach each concept.
The document is a thesis that investigates rational points on elliptic curves and computing their rank. It begins with introductions to algebraic geometry concepts like affine and projective spaces. It defines elliptic curves and discusses their group structure. It explores points of finite order on elliptic curves and relates them to the curve's discriminant. Later sections analyze the group of rational points using Mordell's theorem and descent methods. It also examines specific problems like congruent numbers and constructing elliptic curves with high rank. The thesis provides mathematical foundations and examples to study rational points and ranks of elliptic curves.
1. This document provides information about fundamental chemistry concepts such as atoms, elements, compounds, and chemical reactions. It includes diagrams of atoms and molecules, word equations for chemical reactions, and questions testing understanding of concepts like ions, bonding, and stoichiometry.
2. Questions ask students to identify atoms and molecules in diagrams, explain why some substances shown are elements and others are compounds, name molecules based on their constituent atoms, write word equations for chemical reactions, and perform stoichiometric calculations involving masses of reactants and products.
3. The document provides background information to help students answer questions that assess foundational understanding of atomic structure, the differences between elements and compounds, chemical bonding and reactions, and stoichiometric
This document contains a worksheet with questions about waves. It asks students to identify parts of waves, draw sine waves, order waves by wavelength, amplitude and frequency. It also contains questions calculating wavelength, frequency and velocity given information like speed of sound or light, wavelength or frequency. Graphs are provided showing displacement over time and distance which students are to analyze to determine properties of the waves like wavelength, period, frequency and amplitude.
Worksheet - Refraction in difference mediumNeed Ntk
This document describes an experiment to investigate refraction through a semicircular block with a different refractive index than air. Students are instructed to take angle of incidence and refraction measurements for light passing from air into the block. They will use these measurements to generate graphs of the sine of the angles and determine the refractive index. Comparing this value to the known refractive index allows them to verify Snell's law. The document also prompts students to apply their understanding by relating the results back to the original problem of why pencils appear separated in water and discussing other examples of refraction in daily life.
When light passes from one medium to another, such as from air to glass, it changes direction. This bending of light is called refraction. Refraction occurs because light slows down when moving into a denser medium like glass or water. The denser the second medium, the greater the bending of light. If light travels from a less dense to a more dense medium, it bends toward the normal line. If it travels from more dense to less dense, it bends away from the normal line. This bending allows for phenomena like mirages and the functioning of lenses and prisms.
This document discusses the phenomenon of refraction of light. It begins by explaining that light travels in a straight line through air but can be reflected or refracted at the interface between two substances. When light passes from one medium to another of different density, it changes speed and bends. This bending is called refraction. The direction of bending depends on whether the light is entering a less or more dense medium. Refraction allows lenses and prisms to bend light in useful ways. The document concludes with descriptions of two hands-on activities to demonstrate refraction.
This document outlines three physics experiments on light reflection. Experiment 1 examines diffuse and regular (specular) reflection using a mirror and matte paper. Experiment 2 demonstrates the laws of reflection using a mirror and graduated disk to measure angles of incidence and reflection. It shows that the angle of incidence equals the angle of reflection, and that the incident, reflected, and normal rays lie in the same plane. Experiment 3 reverses the path of light to show the reversibility of reflection - the path of light does not change if the direction of travel is reversed.
This document provides an overview of key concepts in geometry related to circles such as identifying segments and lines that intersect circles, using properties of tangents to solve problems, finding measures of arcs and angles, writing equations of circles, and using coordinate proofs involving circles. Examples and practice problems are included to help teach each concept.
The document is a thesis that investigates rational points on elliptic curves and computing their rank. It begins with introductions to algebraic geometry concepts like affine and projective spaces. It defines elliptic curves and discusses their group structure. It explores points of finite order on elliptic curves and relates them to the curve's discriminant. Later sections analyze the group of rational points using Mordell's theorem and descent methods. It also examines specific problems like congruent numbers and constructing elliptic curves with high rank. The thesis provides mathematical foundations and examples to study rational points and ranks of elliptic curves.
1. This document provides information about fundamental chemistry concepts such as atoms, elements, compounds, and chemical reactions. It includes diagrams of atoms and molecules, word equations for chemical reactions, and questions testing understanding of concepts like ions, bonding, and stoichiometry.
2. Questions ask students to identify atoms and molecules in diagrams, explain why some substances shown are elements and others are compounds, name molecules based on their constituent atoms, write word equations for chemical reactions, and perform stoichiometric calculations involving masses of reactants and products.
3. The document provides background information to help students answer questions that assess foundational understanding of atomic structure, the differences between elements and compounds, chemical bonding and reactions, and stoichiometric
This document contains a worksheet with questions about waves. It asks students to identify parts of waves, draw sine waves, order waves by wavelength, amplitude and frequency. It also contains questions calculating wavelength, frequency and velocity given information like speed of sound or light, wavelength or frequency. Graphs are provided showing displacement over time and distance which students are to analyze to determine properties of the waves like wavelength, period, frequency and amplitude.
Worksheet - Refraction in difference mediumNeed Ntk
This document describes an experiment to investigate refraction through a semicircular block with a different refractive index than air. Students are instructed to take angle of incidence and refraction measurements for light passing from air into the block. They will use these measurements to generate graphs of the sine of the angles and determine the refractive index. Comparing this value to the known refractive index allows them to verify Snell's law. The document also prompts students to apply their understanding by relating the results back to the original problem of why pencils appear separated in water and discussing other examples of refraction in daily life.
The document discusses ratios and proportions involving geometric shapes and quantities. It provides examples of ratios between various attributes of squares, rectangles, triangles, and circles. It also gives practice problems involving expressing ratios in simplest form, finding missing values given ratio relationships, and applications involving rates and proportional reasoning. The focus is on understanding and working with ratios, proportions, and their applications in mathematics.
The document discusses ratios and proportions. It provides examples of ratios between geometric shapes like squares, rectangles, triangles, and circles. It also gives practice problems involving expressing ratios in simplest form, finding missing terms in ratios, and applications involving rates and proportions. The document is a study guide for a math unit on ratios and proportions.
Faster than a..._reading_q_booklet_-_with_sourcessparkly
KS4
Unit 1: Non-Fiction
Unit 1: Non-Fiction
1) This document outlines a reading comprehension exam for a non-fiction unit. It provides three sources related to extreme sports and asks students to answer questions about each source.
2) The questions require students to summarize details about Felix Baumgartner's skydive from Source 1, explain how the headline and picture in Source 2 are effective, and discuss Usain Bolt's thoughts and feelings about his career from Source 3.
3) The final question asks students to compare the language used in Sources 1 and 3 and analyze the different effects created in each text. However, the full sources are not provided so
This document discusses the unit circle and circular functions. It begins by explaining how the unit circle is used to define trigonometric function values and determine the measure of an angle based on its coordinates. It then defines the circular functions in terms of the unit circle and provides examples of evaluating circular function values both numerically and exactly. The document concludes by explaining linear and angular speed for a point rotating along a circle.
11 5 circumfrence and area of a circle lessongwilson8786
This document discusses circumference and area of circles. It defines circumference as the distance around a circle and provides the formulas:
C=2πr (circumference formula)
C=πd (circumference formula using diameter)
It also defines area of a circle as the space enclosed inside the circle and provides the formula:
A=πr^2 (area of a circle formula)
The document presents examples of using these formulas to calculate circumference and area and illustrates their relationships through diagrams and explanations.
The pst-euclide package allows drawing of geometric figures in LaTeX using macros that specify mathematical constraints. It defines points that can be used to construct figures through common transformations and intersections. Basic objects like points, lines, circles, arcs, and curves can be drawn. The package also includes macros for geometric transformations, intersections of objects, and other special objects like midpoints, bisectors, and centers of triangles.
This document discusses different geometric shapes related to circles such as tangents, secants, segments, and sectors. It defines these terms and provides examples of how to calculate angles and areas related to these shapes. Specifically, it defines a tangent as a line intersecting a circle at one point, a secant as a line extending from the circle center to a tangent, and a segment as the region between a chord and its arc. It also provides theorems for calculating angles formed by secants and tangents and shows how to calculate the area of a sector as a proportion of the full circle and the area of a segment by subtracting a triangle area from its sector area.
Lines of Latitude and Longitude – WorksheetYaryalitsa
WORKSHEET on Lines of Latitude, Lines of Longitude, Climate Zones, Equinoxes, Solstices, The Three Norths, Prime Meridian, International Date Line, Greenwich Mean Time, Coordinated Universal Time.
WORKSHEET to work with: Lines of Latitude and Longitude – PowerPoint at:
http://www.slideshare.net/yaryalitsa/lines-of-latitude-and-longitude-powerpoint
The document defines key terms related to circles such as central angle, arc, semicircle, minor arc, and major arc. It provides examples of identifying these elements in circles and finding the measures of central angles and arcs. The document concludes with homework problems involving calculating measures of central angles and arcs in various circle graphs and diagrams.
The document defines key terms related to circles such as central angle, arc, semicircle, minor arc, and major arc. It provides examples of identifying these elements in circles and finding the measures of central angles and arcs. The document also includes practice problems asking students to find the measures of central angles, arcs, and sectors in circles and a circle graph.
This document provides an overview of best practices for data center design. It discusses principles like green design, virtualization, security, and business continuity. It then gives a detailed example data center design covering components like cabling, fire suppression, HVAC, power, racks, and monitoring. The design aims to be efficient, secure, and reliable through features like access control, water detection, and labeling of infrastructure.
This thesis investigates noise from a lift-offset coaxial helicopter configuration using computational modeling. Key findings include:
- Constructive and destructive interference occurs for coaxial thickness noise depending on observer location.
- Blade crossings and blade-vortex interactions are important sources of coaxial loading noise.
- Coaxial thickness noise is generally lower than a single rotor helicopter, especially at forward flight speeds.
- Reducing rotor tip speed through lower RPM provides significant noise reduction potential for coaxial rotors.
- Alternative blade designs like dual-swept tips and curved sweeping can further reduce coaxial thickness noise.
Chord of a Circle Definition Formula Theorem & Examples.pdfChloe Cheney
Finding the chord of a circle is one of the most important terms in math. Learn what the chord of the circle is and how to use the chord length formula with examples and theorems.
This document summarizes key terms and theorems related to circles:
1. It defines circles and related terms like radius, diameter, chord, arc, and sector.
2. It describes theorems like equal chords subtend equal angles at the center, and conversely if angles are equal then chords are equal.
3. Other concepts covered include perpendiculars from the center bisect chords, congruent arcs subtend equal angles, and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Some properties of tangents, secants and chords, Angles formed by intersecting chords, tangent and chord and two secants, Chords and their arcs, Segments of chords secants and tangents, Lengths of arcs and areas of sectors
This dissertation explores strategies for culturally-appropriate urban design and consultation in First Nation communities. It acknowledges the history of discrepancies between Indigenous culture and Western planning approaches. Through a case study of the Blackfoot community of Brocket, Alberta, the author proposes guidelines for urban design that consider both cultural traditions and sustainability. Key aspects discussed include community structure, housing patterns, movement networks, public spaces, and incorporating local context and customs into the design process. The author also presents a "Blackfoot Circle Structure" model for consultation involving focus groups, taskforces, and public meetings to foster community input and support for decisions in a culturally meaningful way.
The document provides information about circles including definitions, properties, theorems and history. It defines a circle as a simple closed curve where all points are equidistant from the center. Key properties discussed are that a circle's circumference and radius are proportional, and its area is proportional to the square of the radius. Theorems covered relate to chords, tangents, secants and inscribed angles. The document also discusses squaring the circle problem and circles in history from ancient Greeks to modern mathematics.
The document provides an introduction to graph theory concepts. It defines graphs as pairs of vertices and edges, and introduces basic graph terminology like order, size, adjacency, and isomorphism. Graphs can be represented geometrically by drawing vertices as points and edges as lines between them. Both simple graphs and multigraphs are discussed.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
The document discusses ratios and proportions involving geometric shapes and quantities. It provides examples of ratios between various attributes of squares, rectangles, triangles, and circles. It also gives practice problems involving expressing ratios in simplest form, finding missing values given ratio relationships, and applications involving rates and proportional reasoning. The focus is on understanding and working with ratios, proportions, and their applications in mathematics.
The document discusses ratios and proportions. It provides examples of ratios between geometric shapes like squares, rectangles, triangles, and circles. It also gives practice problems involving expressing ratios in simplest form, finding missing terms in ratios, and applications involving rates and proportions. The document is a study guide for a math unit on ratios and proportions.
Faster than a..._reading_q_booklet_-_with_sourcessparkly
KS4
Unit 1: Non-Fiction
Unit 1: Non-Fiction
1) This document outlines a reading comprehension exam for a non-fiction unit. It provides three sources related to extreme sports and asks students to answer questions about each source.
2) The questions require students to summarize details about Felix Baumgartner's skydive from Source 1, explain how the headline and picture in Source 2 are effective, and discuss Usain Bolt's thoughts and feelings about his career from Source 3.
3) The final question asks students to compare the language used in Sources 1 and 3 and analyze the different effects created in each text. However, the full sources are not provided so
This document discusses the unit circle and circular functions. It begins by explaining how the unit circle is used to define trigonometric function values and determine the measure of an angle based on its coordinates. It then defines the circular functions in terms of the unit circle and provides examples of evaluating circular function values both numerically and exactly. The document concludes by explaining linear and angular speed for a point rotating along a circle.
11 5 circumfrence and area of a circle lessongwilson8786
This document discusses circumference and area of circles. It defines circumference as the distance around a circle and provides the formulas:
C=2πr (circumference formula)
C=πd (circumference formula using diameter)
It also defines area of a circle as the space enclosed inside the circle and provides the formula:
A=πr^2 (area of a circle formula)
The document presents examples of using these formulas to calculate circumference and area and illustrates their relationships through diagrams and explanations.
The pst-euclide package allows drawing of geometric figures in LaTeX using macros that specify mathematical constraints. It defines points that can be used to construct figures through common transformations and intersections. Basic objects like points, lines, circles, arcs, and curves can be drawn. The package also includes macros for geometric transformations, intersections of objects, and other special objects like midpoints, bisectors, and centers of triangles.
This document discusses different geometric shapes related to circles such as tangents, secants, segments, and sectors. It defines these terms and provides examples of how to calculate angles and areas related to these shapes. Specifically, it defines a tangent as a line intersecting a circle at one point, a secant as a line extending from the circle center to a tangent, and a segment as the region between a chord and its arc. It also provides theorems for calculating angles formed by secants and tangents and shows how to calculate the area of a sector as a proportion of the full circle and the area of a segment by subtracting a triangle area from its sector area.
Lines of Latitude and Longitude – WorksheetYaryalitsa
WORKSHEET on Lines of Latitude, Lines of Longitude, Climate Zones, Equinoxes, Solstices, The Three Norths, Prime Meridian, International Date Line, Greenwich Mean Time, Coordinated Universal Time.
WORKSHEET to work with: Lines of Latitude and Longitude – PowerPoint at:
http://www.slideshare.net/yaryalitsa/lines-of-latitude-and-longitude-powerpoint
The document defines key terms related to circles such as central angle, arc, semicircle, minor arc, and major arc. It provides examples of identifying these elements in circles and finding the measures of central angles and arcs. The document concludes with homework problems involving calculating measures of central angles and arcs in various circle graphs and diagrams.
The document defines key terms related to circles such as central angle, arc, semicircle, minor arc, and major arc. It provides examples of identifying these elements in circles and finding the measures of central angles and arcs. The document also includes practice problems asking students to find the measures of central angles, arcs, and sectors in circles and a circle graph.
This document provides an overview of best practices for data center design. It discusses principles like green design, virtualization, security, and business continuity. It then gives a detailed example data center design covering components like cabling, fire suppression, HVAC, power, racks, and monitoring. The design aims to be efficient, secure, and reliable through features like access control, water detection, and labeling of infrastructure.
This thesis investigates noise from a lift-offset coaxial helicopter configuration using computational modeling. Key findings include:
- Constructive and destructive interference occurs for coaxial thickness noise depending on observer location.
- Blade crossings and blade-vortex interactions are important sources of coaxial loading noise.
- Coaxial thickness noise is generally lower than a single rotor helicopter, especially at forward flight speeds.
- Reducing rotor tip speed through lower RPM provides significant noise reduction potential for coaxial rotors.
- Alternative blade designs like dual-swept tips and curved sweeping can further reduce coaxial thickness noise.
Chord of a Circle Definition Formula Theorem & Examples.pdfChloe Cheney
Finding the chord of a circle is one of the most important terms in math. Learn what the chord of the circle is and how to use the chord length formula with examples and theorems.
This document summarizes key terms and theorems related to circles:
1. It defines circles and related terms like radius, diameter, chord, arc, and sector.
2. It describes theorems like equal chords subtend equal angles at the center, and conversely if angles are equal then chords are equal.
3. Other concepts covered include perpendiculars from the center bisect chords, congruent arcs subtend equal angles, and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Some properties of tangents, secants and chords, Angles formed by intersecting chords, tangent and chord and two secants, Chords and their arcs, Segments of chords secants and tangents, Lengths of arcs and areas of sectors
This dissertation explores strategies for culturally-appropriate urban design and consultation in First Nation communities. It acknowledges the history of discrepancies between Indigenous culture and Western planning approaches. Through a case study of the Blackfoot community of Brocket, Alberta, the author proposes guidelines for urban design that consider both cultural traditions and sustainability. Key aspects discussed include community structure, housing patterns, movement networks, public spaces, and incorporating local context and customs into the design process. The author also presents a "Blackfoot Circle Structure" model for consultation involving focus groups, taskforces, and public meetings to foster community input and support for decisions in a culturally meaningful way.
The document provides information about circles including definitions, properties, theorems and history. It defines a circle as a simple closed curve where all points are equidistant from the center. Key properties discussed are that a circle's circumference and radius are proportional, and its area is proportional to the square of the radius. Theorems covered relate to chords, tangents, secants and inscribed angles. The document also discusses squaring the circle problem and circles in history from ancient Greeks to modern mathematics.
The document provides an introduction to graph theory concepts. It defines graphs as pairs of vertices and edges, and introduces basic graph terminology like order, size, adjacency, and isomorphism. Graphs can be represented geometrically by drawing vertices as points and edges as lines between them. Both simple graphs and multigraphs are discussed.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Assessment and Planning in Educational technology.pptxKavitha Krishnan
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
2. 1. Definition of Circle
A circle is a plane figure bounded by one curved line, and
such that all straight lines drawn from a certain point within
it to the bounding line, are equal. The bounding line is called
its circumference and the point, its centre.
— Euclid, Elements, Book I
3. 2. PART of Circle
1. “Center” is the center of circle
2. “Radius” is the distance from the center to
…..the circumference
3. “Diameter ” is the width of the circle that passesasse
…..through the center
4. “Circumference” is the distance around the edge
…..of a circle.
5. “Arc” is a fraction of the circumference.
1
-
-
-
L
4. 2. PART of Circle
6. “Chord” is a line joining two points on the circumference.
7. “ Secant” is an extended chord that cuts the circle at
……two distinct points.
8. “Tangent” is A line that touches the circumference of
…..a circle at a point.
9. “Sector” is a region bounded by two radii of equal
…..length with a common center.
10. “Segment” is the segment of a circle is the region
…..bounded by a chord and the arc subtended by
…..the chord.
L
5. Semicircle
Major Arc
Minor Arc
Central Angle Inscribed Angle Angle Inscribed in a semicircle
1. Relations of Central Angle, Arcs and Cords
3. Properties of Circle
• •
•
L
6. In one circle, If two arcs are equal, then their corresponding
central angles are equal, and their corresponding chords are
also equal.
arc
c
h
o
r
d
central angle
(Theorem 1)
1. Relations of Central Angle, Arcs and Cords
3. Properties of Circle
1
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7. In one circle, If two arcs are equal, then their corresponding
central angles are equal, and their corresponding chords are
also equal.
(Theorem 1)
Example 1
arc length = 4
A
B
C
D
o If AC = CD and 1 = 45. Find the measure of 2
Textbook-Example 1 (Page 158)
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..…………………………………………………………………………
3. Properties of Circle
1. Relations of Central Angle, Arcs and Cords
1ำ
8. In one circle, If two arcs are equal, then their corresponding
central angles are equal, and their corresponding chords are
also equal.
(Theorem 1)
Example 1
arc length = 4
A B
C
D
o 2. If AB is the diameter, BC = CD = DE and BOC = 40.
Find the measure of AOE
Textbook-Practice (Page 159)
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3. Properties of Circle
1. Relations of Central Angle, Arcs and Cords
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9. 3. Properties of Circle
2. Pythagorean Theorem in Calculating the Arcs
Example 1
2. If OE l AB, the radius is 5, and OE = 3. Find the length of chord AB
Textbook-Example 1 (Page 159)
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10. 3. Properties of Circle
2. Pythagorean Theorem in Calculating the Arcs
Example 1
2. If the radius of circle O is 2 cm., the length of chord AB is 2 cm.
Find the measure of AOB and the distance from O to AB
Textbook-Example 2 (Page 160)
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
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A
C
B
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i = %
i ± s
i %
11. 3. Properties of Circle
2. Pythagorean Theorem in Calculating the Arcs
Example 1
1. If the radius of circle O is 13., the length of chord AB is 24 cm.
Find the distance from O to AB
Textbook-Practice (Page 160)
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
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A B
O
•
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s
12. 3. Properties of Circle
2. Pythagorean Theorem in Calculating the Arcs
Example 1
2. If AB is the diameter of circle O. Chord CD perpendicular
bisects OB at E, CD = 4/3. Find the radius.
Textbook-Practice (Page 160)
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
..……………………………………………………………………………….……
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C
B
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A
•
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13. 3. Properties of Circle
2. Pythagorean Theorem in Calculating the Arcs
Example 1
3. Given the radius of circle O is 20 cm. AB is a chord in circle O,
and AOB =. 120
Textbook-Practice (Page 160)
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14. The angle formed by two line segment in (2) is call circumferential angle.
3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
(1) (2) (3) (4)
15. The circumferential angles corresponding to a semicircle
or the diameter are all equal, which is a right angle, 90 .
The arc that a 90 circumferential angle corresponds to is
the diameter
(Theorem 2)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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A
C
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16. o
If line segment AB is the diameter of circle O. Point C is on circle. Then
ACB is a circumferential angle formed by the diameter AB. What kind
of angle could ACB be?
Textbook-(Page 161)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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17. In one circle, the measures of any circumferential angle
of the same arc are equal and is one half of the measure
of the central angle of that arc
(Theorem 3)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
B
A
C
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18. If AB is the diameter of circle O, and A = 80 . Find the measure of ABC
Textbook- Example 1 (Page 162)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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19. Given AB is the diameter of circle O, and D = 40 . Find the measure of
CAB
Textbook- Example 2 (Page 163)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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20. Example 1
1. Given A, B, and C are points on circle O. ACB is a major arc.
Which of the following has the same measure AOB
A. 2C B. 4B C. 4A D. B + C
Textbook-Practice (Page 163)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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21. Example 1
2. The vertices of ABC, A, B, C are all on circle O.
If ABC + AOC = 90, then AOC =
Textbook-Practice (Page 163)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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22. Example 1
3. The Diameter of circle O, AB = 2, chord AC = 1. Point D is on
circle O, then D =
Textbook-Practice (Page 164)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
o
B
A
C
D
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23. Example 1
3. The Diameter of circle O, AB = 2, chord AC = 1. Point D is on
circle O, then D =
Textbook-Practice (Page 164)
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3. Properties of Circle
3. Circumference Angles (A. Properties of Circumferential Angle)
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