温馨提示：本站仅提供公开网络链接索引服务，不存储、不篡改任何第三方内容，所有内容版权归原作者所有
AI智能索引来源：http://www.edu.com/math-glossary/Circle-Theorems-Definition-Examples
点击访问原文链接

Circle Theorems: Definition and Examples | EDU.COM

进入原网站 导航首页
Circle Theorems: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Circle TheoremsCircle Theorems: Definition and ExamplesTable of ContentsCircle Theorems: Understanding Angles and Relationships in Circles Definition of Circle Theorems

Circle theorems in geometry are statements that prove significant results about circles. These theorems provide important information about various parts of a circle and help us calculate missing angles using established rules. A circle is a locus of points that are equidistant from a fixed point called the center, and different theorems relate to various circle parts such as radius, central angles, tangents, sectors, and chords.

Circle theorems include the alternate segment theorem, the angle at the center theorem, angles in the same segment theorem, angle in a semicircle theorem, chord of a circle theorem, angles subtended by equal chords theorem, cyclic quadrilateral theorem, and tangent theorems. Each theorem establishes specific relationships between angles and other elements of circles, making it easier to solve complex geometric problems involving circles.

Examples of Circle Theorems Example 1: Finding Angles Using the Alternate Segment Theorem Problem:

In the circle given below, triangle ABCABCABC is inscribed in the circle and the tangent DEDEDE meets the circle at the point BBB. Find the measure of angle "xxx" and "yyy".

circle

Step-by-step solution:

Step 1, Find angle xxx using the sum of interior angles in a triangle. We know that the sum of interior angles of a triangle is equal to 180°180°180°.

Step 2, Apply this to triangle ABCABCABC.

∠BAC+∠ACB+∠ABC=180°\angle BAC + \angle ACB + \angle ABC = 180°∠BAC+∠ACB+∠ABC=180°

x+57°+48°=180°x + 57° + 48° = 180°x+57°+48°=180°

Step 3, Solve for xxx.

x=180°−105°x = 180° - 105°x=180°−105°

x=75°x = 75°x=75°

Step 4, Find angle yyy using the alternate segment theorem. According to this theorem, the angle formed between the tangent and the chord through the point of contact equals the angle formed by the same chord in the alternate segment.

Step 5, Apply the theorem to get yyy.

x=y=75°x = y = 75°x=y=75°

Step 6, State the answer. The measure of ∠x∠x∠x and ∠y∠y∠y is 75°75°75°.

Example 2: Using the Angle in a Semicircle Theorem Problem:

In the figure given below, find the value of xxx using the circle theorems.

circle

Step-by-step solution:

Step 1, Identify that ∠ABC∠ABC∠ABC is an angle in a semicircle.

Step 2, Apply the angle in a semicircle theorem. An angle in a semicircle is always a right angle.

m∠ABC=90°m\angle ABC = 90°m∠ABC=90°

Step 3, Use the sum of interior angles of a triangle to find xxx.

m∠BAC+m∠ACB+m∠ABC=180°m\angle BAC + m\angle ACB + m\angle ABC = 180°m∠BAC+m∠ACB+m∠ABC=180°

Step 4, Substitute the known values.

x+37°+90°=180°x + 37° + 90° = 180°x+37°+90°=180°

Step 5, Solve for xxx.

x+127°=180°x + 127° = 180°x+127°=180°

x=180°−127°=53°x = 180° - 127° = 53°x=180°−127°=53°

Step 6, State the answer. The value of xxx is 53°53°53°.

Example 3: Finding the Length of a Chord Problem:

In the given figure, the point OOO is a center of a circle, the radius of a circle is 171717 inches and OP=8OP = 8OP=8 inches. Find the length of the chord ABABAB.

circle

Step-by-step solution:

Step 1, Identify the given information.

OA=OB=17 inchesOA = OB = 17 \text{ inches}OA=OB=17 inches (radius)

OP=8 inchesOP = 8 \text{ inches}OP=8 inches

OP⊥ABOP \perp ABOP⊥AB (perpendicular)

Step 2, Use the Pythagorean theorem in triangle OPBOPBOPB.

OP2+PB2=OB2OP^2 + PB^2 = OB^2OP2+PB2=OB2

82+PB2=1728^2 + PB^2 = 17^282+PB2=172

Step 3, Solve for PBPBPB.

PB2=172−82=289−64=225PB^2 = 17^2 - 8^2 = 289 - 64 = 225PB2=172−82=289−64=225

PB=225=15 inchesPB = \sqrt{225} = 15 \text{ inches}PB=225​=15 inches

Step 4, Apply the chord of a circle theorem. The perpendicular drawn from the center of the circle to a chord bisects the chord.

Step 5, Since OPOPOP is perpendicular to ABABAB, PPP is the midpoint of ABABAB.

AP=PB=15 inchesAP = PB = 15 \text{ inches}AP=PB=15 inches

Step 6, Find the total length of chord ABABAB.

AB=AP+PB=15 inches+15 inches=30 inchesAB = AP + PB = 15 \text{ inches} + 15 \text{ inches} = 30 \text{ inches}AB=AP+PB=15 inches+15 inches=30 inches

Step 7, State the answer. The length of the chord ABABAB is 303030 inches.

Comments(1)HHunterGinaNovember 4, 2025I've used this page to teach circle theorems. The clear defs and examples made it easy for my students to grasp. Great resource!

Explore More TermsClassifyCompatible NumbersDozenPartial QuotientEndpointParallelepipedView All Math TermsRecommended Interactive LessonsUse the Number Line to Round Numbers to the Nearest Ten3Math3.NBT.A.1Multiply by 63Math3.OA.C.7One-Step Word Problems: Division3Math3.OA.D.8Multiply by 43Math3.OA.C.7Identify and Describe Addition Patterns3Math3.OA.D.9Write Multiplication and Division Fact Families3Math3.OA.A.4, 3.OA.B.6View All Interactive LessonsRecommended VideosOdd And Even Numbers2Math2.OA.C.3Convert Units of Mass4Math4.MD.A.1，4.MD.A.2Compare Fractions by Multiplying and Dividing4Math4.NF.A.2Word problems: convert units5Math5.MD.A.1Area of Parallelograms6Math6.G.A.1Use Models and Rules to Divide Fractions by Fractions Or Whole Numbers6Math6.NS.A.1View All VideosRecommended WorksheetsClassify Quadrilaterals Using Shared Attributes3Math3.G.A.1Word problems: addition and subtraction of decimals5Math5.NBT.B.7Compare decimals to thousandths5Math5.NBT.A.3bPercents And Decimals6Math6.RP.A.3Least Common Multiples6Math6.NS.B.4Solve Percent Problems6Math6.RP.A.3cView All WorksheetsRecommended Coloring PagesPrincess holding a bubble wand with bubbles above her headPre-K – KAll SubjectsA cylinder standing on a flat surface with a circle shadow1 – 2All SubjectsA centerpiece with a basket of apples and a small scarecrow decoration1 – 2All SubjectsShantungosaurus walking through a simple forest1 – 2All SubjectsA bowl of ramen with a detailed kitchen counter in the background3 – 4All SubjectsFull moon with a detailed cityscape below5 – 6All SubjectsView All Coloring PagesRecommended BlogsEye-Opening Summer Slide Statistics Every Parent Should KnowOctober 12, 2025A Parent's Guide to the Benefits of Tutoring for Elementary StudentsOctober 11, 2025How to Change ABCya from Spanish to English: A Parent's GuideOctober 5, 2025Top Sites Like ABCya for Kids' Digital Learning FunOctober 4, 2025Age-Appropriate Chores for 10 Year Olds: Building Independence and Life SkillsSeptember 17, 2025The Best Tablet for Autistic Children: A Parent and Teacher's Guide to Making Smart ChoicesSeptember 17, 2025View All Blog PostsQUICK LINKSAbout UsPrivacy PolicyTerms of ServiceTOOLSHomework HelperGuide DesignerPodcast MakerPlan BuilderRESOURCESMath GlossaryEnglish GlossaryEnglish Language ArtsMathematicsScienceBook InsightsFun with WordsBlog© 2025 EDU.COM. All rights reserved.

智能索引记录