Cyclic quadrilateral

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Learning objectives

 * 1) A quadrilateral ABCD is called cyclic if all of its four vertices lie on a circle.
 * 2) In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.
 * 3) If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.
 * 4) In a cyclic quadrilateral the exterior angle is equal to interior opposite angle.

Activity#1 Cyclic quadrilateral
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Developmental Questions:
 * Estimated Time 10 minutes
 * Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
 * Prerequisites/Instructions, if any
 * 1) The students should know a circle and its parts.
 * 2) They should know that a quadrilateral is a 4 sided closed figure.
 * Multimedia resources : Laptop
 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 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.
 * 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 ?

Activity No # 2.Properties of a Cyclic quadrilateral
coloured paper, pair of scissors, sketch pen, carbon paper, geometry box This activity has been taken from the website http://mykhmsmathclass.blogspot.in/2007/11/class-ix-activity-16.html
 * Estimated Time: 45 minutes
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * 1) The students should know a circle and a quadrilateral.
 * 2) They should know that in a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.
 * 3) In a cyclic quadrilateral the exterior angle is equal to interior opposite angle
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets
 * Process:

Developmental Questions: Name the two properties of cyclic quarilaterals.
 * 1) Draw a circle of any radius on a coloured paper and cut it.
 * 2) Paste the circle cut out on a rectangular sheet of paper.
 * 3) By paper folding get chords AB, BC, CD and DA in order.
 * 4) Draw AB, BC, CD & DA. A cyclic quadrilateral ABCD is obtained.
 * 5) Produce AB to form a ray AE such that exterior angle CBE is formed.
 * 6) Make a replica of cyclic quadrilateral ABCD using carbon paper.
 * 7) Cut the replica into 4 parts such that each part contains one angle.
 * 8) Draw a straight line on a paper.
 * 9) Place the two opposite angles, angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
 * 10) Place other two opposite angles, angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
 * 11) Make a replica of angle ADC and place it on angle CBE . Write the observation.
 * 1) How do you take radius ?
 * 2) What is the circumference ?
 * 3) What is a chord ?
 * 4) What is a quadrilateral ?
 * 5) Where are all four vertices of a quadrilateral located ?
 * 6) What part are we trying to cut and compare ?
 * 7) What can you infer ?
 * Evaluation:
 * 1) Angle BAD and angle BCD, when placed adjacent to each other on a straight line, completely cover the straight angle.What does this mean ?
 * 2) Angle ABC and angle ADC, when placed adjacent to each other on a straight line, completely cover the straight angle.What can you conclude ?
 * 3) Compare angle ADC with angle CBE.
 * Question Corner:

Learning objectives

 * 1) Ability to construct a cyclic quadrilateral accurately.

Activity No # Constructing a cyclic quadrilateral

 * Estimated Time: 40 minutes.
 * Materials/ Resources needed:
 * 1) Laptop, geogebra file, projector and a pointer.
 * 2) Students constructing materials, the geometry box.
 * 3) white papers.
 * Prerequisites/Instructions, if any
 * 1) 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:
 * 1) The teacher can do this activity after introducing the concept and properties of cyclic quadrilateral.
 * 2) She can project the file and let students watch it carefully.
 * 3) After watching discuss the steps of construction and the purpose of each step so that the students can appreciate the sequence of construction steps.
 * 4) Then ask the students to actually construct a cyclic quadrilateral for the given measures.
 * Developmental Questions:
 * 1) What is a cyclic quadrilateral ? Why is it called so ?
 * 2) Name the measuring parameters of it ?
 * 3) What measures are given for its construction ?
 * 4) Explain the steps involved in determing the radius of the required circle ?
 * 5) What do the measures of the arcs specify ?
 * Evaluation:
 * 1) Were the students able to justify the sequence of steps involved ?
 * Question Corner:
 * 1) Can you draw a circle first and then the quadrilateral ? Why not so ?

Activity No #

 * Estimated Time
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets
 * Process/ Developmental Questions
 * Evaluation
 * Question Corner

Learning objectives
Converse theorems:
 * 1) Both pairs of opposite angles of a cyclic quadrilateral are supplementary.
 * 2) When one side of a cyclic quadrilateral  is produced, the exterior angle so formed is equal to the interior opposite angle.
 * 1) Suppose a quadrilateral is such that the sum of two opposite angles is a straight angle, them the quadrilateral is cyclic.
 * 2) If the exterior angle of a quadrilateral is equal to the interior opposite angle, then the quadrilateral is cyclic.

Activity No 1. Theorems
Laptop, geogebra file, projector and a pointer. <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" />
 * Estimated Time : 40 minutes.
 * Materials/ Resources needed:
 * Prerequisites/Instructions, if any
 * 1) The students should know a cyclic quadrilateral and its properties.
 * 2) They should know the linear pair and exterior angle theorem.
 * 3) They should know the circle theorem (Angle at centre = double the angle at the circumference)
 * Multimedia resources: Laptop
 * Website interactives/ links/ / Geogebra Applets:
 * Process:
 * 1) The teacher can project the geogebra file and prove the theorems.
 * Developmental Questions:
 * 1) How many angles does a cyclic quadrilateral have ?
 * 2) Name the opposite angles of it.
 * 3) Name the minor arc.
 * 4) Recall the angle -arc theorem.
 * 5) What is the total angle at the centre of a circle ?
 * 6) Name the angles at the centre of the circle.
 * 7) What is the sum of those two angles ?
 * 8) How can you show that <b and <d are supplementary from above observations ?
 * Evaluation;
 * 1) What is the converse of this theorem.
 * Question Corner;
 * 1) Write down the steps to prove the converse of this theorem.

Activity No #

 * 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 =

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