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Geometry - Circles - Document preview page 1

Geometry - Circles - Page 1

Document preview content for Geometry - Circles

Geometry - Circles

This document provides study materials related to Geometry - Circles. It may include explanations, summarized notes, examples, or practice questions designed to help students understand key concepts and review important topics covered in their coursework.

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Geometry - Circles - Page 1 preview imageStudy GuideGeometry–Circles1. Central Angles and ArcsCircles have several different types of angles. The most important one to start with is thecentralangle.Acentral angleis an angle whose vertex (corner point) is at the center of the circle. Its sides are radiiof the circle.A full circle measures360°. That means a central angle can “sweep” across part of the circle, and itsmeasure tells us how much of the circle is covered.Figure 1In the figure,∠AOB is a central angle because:•Point O is the center.•OA and OB are radii.1.1Understanding ArcsAnarcis a part of the circle’s edge.
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Geometry - Circles - Page 2 preview imageStudy GuideIt includes:•Two endpoints•All the points between them along the circleWe use a curved line symbol over the letters to name an arc.There arethree types of arcs:SemicircleAsemicircleis half of a circle.•It measures180°.•Its endpoints are the endpoints of adiameter.•It is named usingthree letters.The first and last letters are the endpoints of the diameter.The middle letter is any point on that arc.Figure 2If AC is a diameter, then arc ABC is a semicircle.Minor ArcAminor arcis smaller than a semicircle.•Its measure isless than 180°.•It is named usingtwo letters(just the endpoints).
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Geometry - Circles - Page 3 preview imageStudy GuideFigure 3Major ArcAmajor arcis larger than a semicircle.•Its measure ismore than 180°.•It is named usingthree letters.The first and last letters are the endpoints.The middle letter is a point somewhere along the arc.Figure 41.2Measuring ArcsArcs can be measured in two ways:1.Degree measure2.Arc length (units like inches or cm)Let’s look at each type.
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Geometry - Circles - Page 4 preview imageStudy GuideDegree Measure of a SemicircleA semicircle always measures:Its length ishalf of the circumference.Degree Measure of a Minor ArcThe measure of a minor arc is thesame as its central angle.Its length is a portion of the circumference and is alwaysless than half.Degree Measure of a Major ArcTo find a major arc:Its length is alwaysmore than halfthe circumference.Important NotationExample 1:SolveCircle O has diameter AB and OB = 6 inches.Find:•(a) the degree measure of arc AXB•(b) the length of arc AXB
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Geometry - Circles - Page 5 preview imageStudy GuideFigure 5Since AXB is a semicircle:To find the length:Circumference formula:Here, radius = 6Full circumference:Half of that:So the arc length is:
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Geometry - Circles - Page 6 preview imageStudy Guide1.3Arc Addition PostulateIf point B is between A and C on a circle, then:This simply means:Small arcs add up to make a larger arc.Example 2:FindIf:Using the Arc Addition Postulate:Figure 6
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Geometry - Circles - Page 7 preview imageStudy GuideExample 3:SolveCircle P has diameter QS.Find:Figure 7Key ideas used:•Minor arc = central angle•Semicircle = 180°•Major arc = 360° − minor arc1.4Theorems About Arcs and Central AnglesTheorem 1:If two central angles are equal, then their corresponding minor arcs are equal.Theorem 2:If two minor arcs are equal, then their corresponding central angles are equal.
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Geometry - Circles - Page 8 preview imageStudy GuideExample 4:SolveCircle O has diameters AC and BD.If ( m∠1 = 40° ), find the remaining measures.Figure 8Key results:•Minor arc equals its central angle.•Vertical angles are equal.•A semicircle measures 180°.•The angles of a triangle add to 180°.After solving:•Some arcs measure 40°•Some measure 140°•∠3 and∠4 both equal 20°Summary• A central angle has its vertex at the center.• Minor arc < 180°• Semicircle = 180°
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Geometry - Circles - Page 9 preview imageStudy Guide• Major arc > 180°• Minor arc measure = central angle• Major arc = 360° − minor arc• Arcs can be added together2. Arcs and Inscribed AnglesSo far, we’ve talked aboutcentral angles—angles whose vertex is at the center of a circle.But central angles are not the only angles found in circles. In fact, many important angles are formedon the circle itself.2.1What Is an Inscribed Angle?Aninscribed angleis:•An angle formed bytwo chords•With itsvertex on the circleThis is different from a central angle, whose vertex is at the center.2.2What Is an Intercepted Arc?Every inscribed angle “cuts off” a part of the circle.That part is called theintercepted arc.It includes:•The endpoints of the angle•The portion of the circle between those endpoints
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Geometry - Circles - Page 10 preview imageStudy GuideFigure 1In this figure:•∠ABC is an inscribed angle.•Arc AC is its intercepted arc.2.3Angles That Are NOT Inscribed AnglesNot every angle inside a circle is inscribed.For an angle to be inscribed:•Its vertex must lieon the circle.•Its sides must bechords.If the vertex is inside the circle but not on it, the angle is NOT inscribed.Figure 2
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Document Details

University
Texas A&M University
Course
Geometry
Subject
Mathematics

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