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Line 4: − + − + − + − + − + − + − + − + − = Concept Map = − [[File:Cyclic_quadrilateral.mm|flash]]+ − − = Textbook = − To add textbook links, please follow these instructions to: − ([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage]) − − =Additional Information= − ==Useful websites== − ==Reference Books== − − = Teaching Outlines = − − ==Concept # 1. Cyclic quadrilateral and its properties== − ===Learning objectives=== − # A quadrilateral ABCD is called cyclic if all of its four vertices lie on a circle. − # In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees. − # If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic. − # In a cyclic quadrilateral the exterior angle is equal to interior opposite angle. − ===Notes for teachers=== Line 48:
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Line 46: − + − + − ===Activity No # 2.Properties of a Cyclic quadrilateral=== + − *Estimated Time: 45 minutes − *Materials/ Resources needed − coloured paper, pair of scissors, sketch pen, carbon paper, geometry box − *Prerequisites/Instructions, if any − # In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees. − # In a cyclic quadrilateral the exterior angle is equal to interior opposite angle − *Multimedia resources − *Website interactives/ links/ / Geogebra Applets − This activity has been taken from the website http://mykhmsmathclass.blogspot.in/2007/11/class-ix-activity-16.html − *Process: − [[File:c.q.jpeg|300px]] − − # Draw a circle of any radius on a coloured paper and cut it. − # Paste the circle cut out on a rectangular sheet of paper. − # By paper folding get chords AB, BC, CD and DA in order. − # Draw AB, BC, CD & DA. A cyclic quadrilateral ABCD is obtained. − # Produce AB to form a ray AE such that exterior angle CBE is formed. − # Make a replica of cyclic quadrilateral ABCD using carbon paper. − # Cut the replica into 4 parts such that each part contains one angle . − # Draw a straight line on a paper. − # Place the two opposite angles, angle BAD and angle BCD adjacent to each other on the straight line.Write the observation. − # Place other two opposite angles, angle ABC and angle ADC adjacent to each other on the straight line . Write the observation. − # Make a replica of angle ADC and place it on angle CBE . Write the observation. − Developmental Questions: − # How do you take radius ? − # What is the circumference ? − # What is a chord ? − # What is a quadrilateral ? − # Where are all four vertices of a quadrilateral located ? − # What part are we trying to cut and compare ? − # What can you infer ? − − # Angle BAD and angle BCD, when placed adjacent to each other on a straight line, completely cover the straight angle.What does this mean ? − # Angle ABC and angle ADC, when placed adjacent to each other on a straight line, completely cover the straight angle.What can you conclude ? − # Compare angle ADC with angle CBE. − *Question Corner: − Name the two properties of cyclic quarilaterals. − − ==Concept # 2.Construction of cyclic quadrilateral== − ===Learning objectives=== − # Ability to construct a cyclic quadrilateral accurately . − ===Notes for teachers=== − ===Activity No # Constructing a cyclic quadrilateral=== − {| style="height:10px; float:right; align:center;" − |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"> − ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div> − |} − *Estimated Time: 40 minutes. − *Materials/ Resources needed: − # Laptop, geogebra file, projector and a pointer. − # Students constructing materials, the geometry box. − # white papers. − *Prerequisites/Instructions, if any − # Sufficient knowledge regarding construction of perpendicular lines, bisectors, angles and circle. − *Multimedia resources : Laptop − *Website interactives/ links/ / Geogebra Applets: For step by step illustration of cyclic quadrilateral construction please refer to the website: http://www.matrusrieppower.net/Constructionoftriangleandcyclicquadrilateral.html. − *Process: − # The teacher can do this activity after introducing the concept and properties of cyclic quadrilateral. − # She can project the file and let students watch it carefully. − # After watching discuss the steps of construction and the purpose of each step so that the students can appreciate the sequence of construction steps. − # Then ask the students to actually construct a cyclic quadrilateral for the given measures. − *Developmental Questions: − # What is a cyclic quadrilateral ? Why is it called so ? − # Name the measuring parameters of it ? − # What measures are given for its construction ? − # Explain the steps involved in determing the radius of the required circle ? − # What do the measures of the arcs specify ? − *Evaluation: − # Were the students able to justify the sequence of steps involved ? − *Question Corner: − # Can you draw a circle first and then the quadrilateral ? Why not so ? − ===Activity No # ===+ − {| style="height:10px; float:right; align:center;" − |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"> − ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div> − |} − *Estimated Time − *Materials/ Resources needed − *Prerequisites/Instructions, if any − *Multimedia resources − *Website interactives/ links/ / Geogebra Applets − *Process/ Developmental Questions − *Evaluation − *Question Corner − ==Concept # 3. Theorems on cyclic quadrilaterals==+ − ===Learning objectives===+ − # Both pairs of opposite angles of a cyclic quadrilateral are supplementary.+ − # When one side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle.<br> − Converse theorems: − # Suppose a quadrilateral is such that the sum of two opposite angles is a straight angle, them the quadrilateral is cyclic. − # If the exterior angle of a quadrilateral is equal to the interior opposite angle, then the quadrilateral is cyclic. − ===Notes for teachers=== − ===Activity No 1. Theorems === − {| style="height:10px; float:right; align:center;" − |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"> − − |} − *Estimated Time : 40 minutes. − *Materials/ Resources needed: − Laptop, geogebra file, projector and a pointer. − *Prerequisites/Instructions, if any − # A cyclic quadrilateral and its properties. − # The linear pair and exterior angle theorem. − # The circle theorem (Angle at centre = double the angle at the circumference) − *Multimedia resources: Laptop − *Website interactives/ links/ / Geogebra Applets: − This geogebra file was done by ITfC-Edu-Team. − <ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> − *Process: − # The teacher can project the geogebra file and prove the theorems. − *Developmental Questions: − # How many angles does a cyclic quadrilateral have ? − # Name the opposite angles of it. − # Name the minor arc. − # Recall the angle -arc theorem. − # What is the total angle at the centre of a circle ? − # Name the angles at the centre of the circle. − # What is the sum of those two angles ? − # How can you show that <b and <d are supplementary from above observations ? − *Evaluation; − # What is the converse of this theorem. − *Question Corner; − # Write down the steps to prove the converse of this theorem. − ===Activity No #=== − {| style="height:10px; float:right; align:center;" − |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"> − ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div> − *Estimated Time − *Materials/ Resources needed − *Prerequisites/Instructions, if any − *Multimedia resources − *Website interactives/ links/ / Geogebra Applets − *Process/ Developmental Questions − *Evaluation − *Question Corner − − = Hints for difficult problems = − − = Project Ideas = − − = Math Fun = − − '''Usage''' − − Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
→Activity No #
{| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;"
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|style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
| style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|
| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
|}
|}
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
__FORCETOC__
__FORCETOC__
===Activity#1 Cyclic quadrilateral ===
===Activity#1 Cyclic quadrilateral ===
*Estimated Time 10 minutes
*Estimated Time 10 minutes
*Multimedia resources : Laptop
*Multimedia resources : Laptop
*Website interactives/ links/ / Geogebra Applets
*Website interactives/ links/ / Geogebra Applets
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<span> </span>
<span></span><div id="ggbContainer5061ec2b58633e074b0c05f395a3849d"></div><span></span>
*Process:
*Process:
# The teacher can recall the concept of a circle, quadrilateral, circumcircle.
# The teacher can recall the concept of a circle, quadrilateral, circumcircle.
*Question Corner
*Question Corner
# Can all quadrilaterals be cyclic ?
# Can all quadrilaterals be cyclic ?
# What are the necessary conditions for a quadrilateral to be cyclic ?
# What are the necessary conditions for a quadrilateral to be cyclic ?
*
*Evaluation:
<span> </span>
<span></span><div id="ggbContainer7ac33d1e68b29e64169259c6189c0d53"></div><span></span>
*
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''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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