[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207778-en":3,"doc-seo-207778-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207778,2336475104957,"นรินทร์","https://ap-avatar.wpscdn.com/avatar/22000c4c6bd8a5076e1?x-image-process=image/resize,m_fixed,w_180,h_180&k=1787554080175789136",4,"Exam","11 Circle Theorems","This document contains a series of geometry problems related to circle theorems, presented as exam-style questions. The problems require students to calculate angles and lengths within circles, applying properties such as tangents, diameters, and angles subtended by arcs. Specific questions involve determining angles using given tangent lines and inscribed angles, calculating lengths using trigonometric ratios in right-angled triangles formed within circles, and proving geometric relationships like the angle at the center being double the angle at the circumference. The document includes diagrams to illustrate the geometric configurations, with numerical values for angles and lengths provided for calculations. Each question is marked with a point value, indicating its weight in an assessment context. The overall structure suggests a non-calculator section of a mathematics examination focusing on circle geometry.","NAME  \nTime:  \nNon-Calculator Questions  \n1 P, Q, R and S are points on the circumference of a circle, centre O. Angle PSR = 73°  \nWork out the size of the angle marked x.  \n(2 marks)  \n3 X and Y are points on the circumference of a circle, centre O.  \nTXis a tangent to the circle.  \nAngle XOY= 68°  \nWork out the size of angle YXT. Give reasons for each stage of your working.  \n(4 marks)  \n5 XZ is a diameter of a circle, centre O.  \nYis a point on the circumference of the circle.  \nWork out the size, in degrees, of angle XZY.  \n(3 marks)  \n7 In the diagram, A, B, C and D are points on the circumference of a circle.  \nXAYis a tangent to the circle.  \nAngle BAY = 63°  \nAngle BCD = 92°  \nWork out the size of the angle marked t.  \nGive a reason for each stage of your working.  \n(3 marks)  \n9 X, Yand Z are points on the circumference of a circle, centre O. UZVis a tangent to the circle.  \nAngle YXZ = 75° and angle XZY= 48°  \nWork out the size of angle XZU.  \nGive reasons for each stage of your working.  \n(3 marks)  \n11 X, Yand Z are points on the circumference of a circle, centre O. XOZ is a diameter of the circle.  \nWO = 10 cm  \nAngle WXO = 90°  \nAngle OWX = 30°  \nAngle YXZ = 30°  \na Explain why angle XYZ is 90°  \n(1 mark)  \nb Calculate the length XY.  \nGive your answer in surd form.  \n(4 marks)  \n13 X, Yand Z are points on the circumference of a circle, centre O.  \nProve that angle YOZ is twice the size of angle YXZ.  \n(4 marks)  \nOverall mark /","cbCaimlH3v3v970p","https://ap.wps.com/l/cbCaimlH3v3v970p","pdf",250811,1,5,"English","en",105,"# Circle Theorems\n## Angles and Lengths Calculations\n## Proofs of Geometric Relationships","[{\"question\":\"What are the key circle theorems tested in this document?\",\"answer\":\"The document tests several circle theorems, including properties of tangents, angles subtended by arcs at the center and circumference, angles in a semicircle, and cyclic quadrilaterals. It also involves applying trigonometry to find lengths within circle-related figures. \"},{\"question\":\"What is the significance of the provided diagrams and markings?\",\"answer\":\"The diagrams illustrate the geometric setups for each problem, and the markings denote given angles, lengths, tangent points, centers, and diameters, which are crucial for solving the problems. Right-angle symbols and angle measures directly guide the application of geometric principles. \"},{\"question\":\"What is expected in terms of problem-solving approach and presentation?\",\"answer\":\"For most questions, students are expected to provide step-by-step working out and give reasons for each stage of their calculation, particularly when dealing with angles. Answers for lengths may need to be in surd form, indicating a need for exact calculations.\"}]","11 Circle Theorems | PDF",1788601111,13,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"11-circle-theorems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/11-circle-theorems/207778/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-06","2026-09-05",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are the key circle theorems tested in this document?","Question",{"text":75,"@type":76},"The document tests several circle theorems, including properties of tangents, angles subtended by arcs at the center and circumference, angles in a semicircle, and cyclic quadrilaterals. It also involves applying trigonometry to find lengths within circle-related figures.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the significance of the provided diagrams and markings?",{"text":80,"@type":76},"The diagrams illustrate the geometric setups for each problem, and the markings denote given angles, lengths, tangent points, centers, and diameters, which are crucial for solving the problems. Right-angle symbols and angle measures directly guide the application of geometric principles.",{"name":82,"@type":73,"acceptedAnswer":83},"What is expected in terms of problem-solving approach and presentation?",{"text":84,"@type":76},"For most questions, students are expected to provide step-by-step working out and give reasons for each stage of their calculation, particularly when dealing with angles. 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