Difference between revisions of "Cyclic quadrilateral"

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A quadrilateral ABCD is called cyclic if all four vertices of it 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.
  
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===Objectives===
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Understanding cyclic quadrilaterals
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[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]
 
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
 
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
 
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
 
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[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
 
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 
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= Concept Map =
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Relation between angles of a cyclic quadrilateral.
__FORCETOC__
 
<mm>[[Cyclic_quadrilateral.mm|flash]]</mm>
 
  
= Textbook =
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===Estimated Time===
To add textbook links, please follow these instructions to:
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10 minutes
([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])
 
  
=Additional Information=
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===Prerequisites/Instructions, prior preparations, if any===
==Useful websites==
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Circle and quadrilaterals should have been introduced.
==Reference Books==
 
  
= Teaching Outlines =
+
===Materials/ Resources needed===
 +
Digital : Laptop, geogebra file, projector and a pointer.
  
==Concept # 1. Cyclic quadrilateral and its properties==
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Geogebra file: [https://ggbm.at/jdxxnrmb Cyclic quadrilateral.ggb]
===Learning objectives===
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# A quadrilateral ABCD is called cyclic if all of its four vertices lie on a circle.
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{{Geogebra|jdxxnrmb}}
# In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.
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# If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.
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===Process (How to do the activity)===
# In a cyclic quadrilateral the exterior angle is equal to interior opposite angle.
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<span></span><span></span>
===Notes for teachers===
 
===Activity#1 Cyclic quadrilateral ===                                                                                                             
 
*Estimated Time 10 minutes
 
*Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
 
*Prerequisites/Instructions, if any
 
# Circle and quadrilaterals should have been introduced.
 
*Multimedia resources : Laptop
 
*Website interactives/ links/ / Geogebra Applets
 
<ggb_applet width="1282" height="601" version="4.0" ggbBase64="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*Process:
 
 
# The teacher can recall the concept of a circle, quadrilateral, circumcircle.
 
# The teacher can recall the concept of a circle, quadrilateral, circumcircle.
 
# Can explain a cyclic quadrilateral and show the geogebra applet.
 
# Can explain a cyclic quadrilateral and show the geogebra applet.
 
# Move points, the vertices of the quadrilateral and let the students observe the sum of opposite interior angles.
 
# Move points, the vertices of the quadrilateral and let the students observe the sum of opposite interior angles.
Developmental Questions:
+
* Developmental Questions:
 
# What two figures do you see in the figure ?
 
# What two figures do you see in the figure ?
 
# Name the vertices of the quadrilateral.
 
# Name the vertices of the quadrilateral.
Line 63: Line 34:
 
*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 ?   <span></span><span></span>
 
 
===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
 
# The students should know a circle and a quadrilateral.
 
# They should know that 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 ?
 
*Evaluation:
 
# 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
 
# The students should have 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>
 
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*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;">
 
''[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.
 
*Prerequisites/Instructions, if any
 
# The students should know a cyclic quadrilateral and its properties.
 
# They should know the linear pair and exterior angle theorem.
 
# They should know the circle theorem (Angle at centre = double the angle at the circumference)
 
*Multimedia resources: Laptop
 
*Website interactives/ links/ / Geogebra Applets:
 
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*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 #===
 
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''[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
+
[[Category:Circles]]

Latest revision as of 05:31, 31 October 2019

A quadrilateral ABCD is called cyclic if all four vertices of it 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.

Objectives

Understanding cyclic quadrilaterals

Relation between angles of a cyclic quadrilateral.

Estimated Time

10 minutes

Prerequisites/Instructions, prior preparations, if any

Circle and quadrilaterals should have been introduced.

Materials/ Resources needed

Digital : Laptop, geogebra file, projector and a pointer.

Geogebra file: Cyclic quadrilateral.ggb


Download this geogebra file from this link.


Process (How to do the activity)

  1. The teacher can recall the concept of a circle, quadrilateral, circumcircle.
  2. Can explain a cyclic quadrilateral and show the geogebra applet.
  3. Move points, the vertices of the quadrilateral and let the students observe the sum of opposite interior angles.
  • Developmental Questions:
  1. What two figures do you see in the figure ?
  2. Name the vertices of the quadrilateral.
  3. Where are all the 4 vertices situated ?
  4. Name the opposite interior angles of the quadrilateral.
  5. What do you observe about them.
  • Evaluation:
  1. Compare the cyclic quadrilateral to circumcircle.
  • Question Corner
  1. Can all quadrilaterals be cyclic ?
  2. What are the necessary conditions for a quadrilateral to be cyclic ?