The longest chord passes through the centre of the circle

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=Additional Information=

Useful websites

 * 1) www.regentsprep.com conatins good objective problems on chords and secants
 * 2) www.mathwarehouse.com contains good content on circles for different classes
 * 3) staff.argyll contains good simulations

Reference Books
= Teaching Outlines = Chord and its related theorems

Learning objectives

 * 1) Meaning of circle and chord.
 * 2) Method to measure the perpendicular distance of the chord from the centre of the circle.
 * 3) Properties of chord.
 * 4) Able to relate chord properties to find unknown measures in a circle.
 * 5) Apply chord properties for proof of further theorems in circles.
 * 6) Understand the meaning of congruent chords.

Notes for teachers

 * 1) A chord is a straight line joining 2 points on the circumference of a circle.
 * 2) Chords within a circle can be related in many ways.
 * 3) The theorems that involve chords of a circle are :
 * Perpendicular bisector of a chord passes through the center of a circle.
 * Congruent chords are equidistant from the center of a circle.
 * If two chords in a circle are congruent, then their intercepted arcs are congruent.
 * If two chords in a circle are congruent, then they determine two central angles that are congruent.

Activity No 1[Theorem 1: Perpendicular bisector of a chord passes through the center of a circle.]
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 * Estimated Time:20 minutes
 * Materials/ Resources needed:Laptop, Geogebra file, projector and a pointer.
 * Prerequisites/Instructions, if any:
 * 1) Basic concepts of a circle and its related terms should have been covered.
 * Multimedia resources: Laptop.
 * Website interactives/ links/ / Geogebra Applets:
 * Process:
 * 1) Show the children the geogebra file and ask the listed questions below.
 * Developmental Questions:
 * 1) What is a chord ?
 * 2) At how many points on the circumference does the chord touch a circle.
 * 3) What is a bisector ?
 * 4) What is a perpendicular bisector ?
 * 5) In each case the perpendicular bisector passes through which point ?
 * Evaluation
 * 1) What is the angle formed at the point of intersection of chord and radius ?
 * 2) Are the students able to understand what a perpendicular bisector is ?
 * 3) Are the students realising that perpendicular bisector drawn for any length of chords for any circle always passes through the center of the circle.
 * Question Corner:
 * 1) What do you infer ?
 * 2) How can you reason that the perpendicular bisector for any length of chord always passes through the centre of the circle.

Activity No # 2.[Theorem 2.Congruent chords are equidistant from the center of a circle.]
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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:Laptop, geogebra,projector and a pointer.
 * Prerequisites/Instructions, if any
 * 1) Basics of circles and its related terms should have been done.
 * Multimedia resources: Laptop, geogebra file, projector and a pointer.
 * Website interactives/ links/ / Geogebra Applets : This geogebra file has been created by Tharanath Achar of Dakshina kannada.
 * Process:
 * 1) Show geogebra file and ask the questions below.
 * Developmental Questions:
 * 1) What is a chord ?
 * 2) Name the centre of the circle.
 * 3) How do you draw congruent chords in a circle ?
 * 4) How many chords do you see in the figure ? Name them.
 * 5) If  both the chords are congruent, what can you say about the length of both the chords ?
 * 6) How can we measure the length of the chord ?
 * 7) What is the procedure to draw perpendicular bisector ?
 * 8) What does theorem 1 say ? Do you all remember ?
 * 9) What is the length of both chords here ?
 * 10) What can you conclude ?
 * 11) Repeat this for circles of different radii and for different lengths of congruent chords.
 * Evaluation:
 * 1) Were the students able to comprehend the drawing of congruent chords in a circle ?
 * 2) Were the students able to comprehend why congruent chords are always equal for a given circle. Let any student explain the analogy.
 * 3) Are the students able to understand that this theorem can be very useful in solving problems related to circles and triangles ?
 * Question Corner:
 * 1) What is a chord ?
 * 2) What are congruent chords ?
 * 3) Why do you think congruent chords are always equal for a circle of given radius ?

Activity No #

 * Estimated Time
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets


 * Process/ Developmental Questions
 * Evaluation
 * Question Corner

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

 * 1) The secant is a line passing through a circle touching it at any two points on the circumference.
 * 2) A tangent is a line toucing the circle at only one point on the circumference.

Activity No # 1.Understanding Secant and Tangent using geogebra.
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Developmental Questions:
 * Estimated Time: 15 minutes
 * Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
 * Prerequisites/Instructions, if any:
 * 1) The students should have a prior knowledge about a circle and its basic parts and terms.
 * 2) They should know the clear distinction between radius, diameter, chord, secant and tangent.
 * Multimedia resources : Laptop and projector
 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 1) The teacher can show the geogebra file.
 * 2) Move the points on circumference and explain secant.
 * 3) When both endpoints of secant meet, it becomes a tangent.
 * 1) Name the points on the circumference of the circle.
 * 2) At how many points is the line touching the circle ?
 * 3) What is the line called ?
 * Evaluation
 * 1) What is the difference between the secant and a tangent?
 * 2) What is the difference between the chord and a secant ?
 * Question Corner
 * 1) Can you draw a secant touching 3 points on the circle ?
 * 2) At how many points does a tangent touch a circle ?
 * 3) How many tangents can be drawn to a circle ?
 * 4) How many tangents can be drawn to a circle at any one given point ?
 * 5) How many parallel tangents can a circle have at the most ?

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

 * 1) The students should know that tangent is a straight line touching the circle at one and only point.
 * 2) They should understand that a tangent is perpendicular to the radius of the circle.
 * 3) The construction protocol of a tangent.
 * 4) Constructing a tangent to a point on the circle.
 * 5) Constructing tangents to a circle from external point at a given distance.
 * 6) A tangent that is common to two circles is called a common tangent.
 * 7) A common tangent with both centres on the same side of the tangent is called a direct common tangent.
 * 8) A common tangent with both centres on either side of the tangent is called a transverse common tangent.

Activity No # 1. Construction of Direct common tangent
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aD8YCH8DMtH4FhXqcMqA1c0BsRJX4gzR71QtEndDVQkdVVVaYrFokFUbTU1rmOxhKjHPMSS5jJwMd5shUS2JNGG1RjobpWACErwRFm7RSqHFOoQXcuVjAFxomc0ZEncISqwu4BcEU7gyd4DaNxS6BSQLuGvZPn6OgQkot+CDoJCwGEv7sqR1kV5obzHC5Y9XtiUuw7hek88GTAbW+t9ULY1FyHpCxG2hSuYVtGbJ3oFO9kazlwydYBQM6vTBSDZcXEodjMmKyM8HIOH9jsNWKwR52wuL7eBgobh0pgQGg9i7tuEbF+fMY8HiU/zwsCVIly5FYfLt/SoObUMvusXw+MJTuigdf8FCxYgOzv7wc2LNaDBilBcVrHwLdyB5Pi2bRj83fP24RDnOrRnz+iBUulJ0ojC5NYqZAiHm17vbh7jiEI5VdDVS7kY6C3E/Pm279mXLVvGCR009IFYxLOXs/HLGUZoaK1RPEd6PWyvA1DeT2uHo/p4mvKN1ml4RhP/ha3R0GlIDrs0cCo2VjMgBWLfWzeu2AyaonHph5qbUSq2U3YTshYbeluRHjY6ovccOeRuAkIJ5iiBA4nKGcL6OUp2/d1333FCBwd9IAbhd7a0tIyyHsoOwdu3b0uAyOPPlR2A6o5Dex2G6g5CdQcizVlIgLWLJvi0yKx4VChNQ+KGytJSlLjRgbjTkotrb5wZgaPSbMaBpkYUJCdJWU3MRoM1s4mqmVUodE1jVhN1ulI5sRzBkT08KSTLfV26dEnqzJOth7JD8OTJk9K2ixcvSp19yg5ArR2Hyg5HddaTctH0bmtrk+KzpgUg83XWDO5aksZt8PNDjfgCcsU/pSWzyd/+4SUkCZPcuqMWZX83IGUxoWwmzobavka+iAtAKMUPWQpKJJduHL0fgVFsMnFFvwdlCB+xsrKSc/OOtEWFLutsJ8Kxmzhu/nxkCzNK/SRb8vIcZjsxh4WhuCQfheUlyK/ZbNOMsqdeoZ0OMivqw+5ai+b69VKqHzU46aJ5kLhuHWd859e8468EnXUKNc2pR4VvkrhmDYrErw31ulNSuV319VJTjMLlKVwlYY03Oo7sxJAwnyVhYU7hoNGDp1W5eWlZURop5buiVD6U0seRVUkSoBYJy8FwMCATJppTsE+jJdGib778Es+eaIJ+0WPYLZy6A8L5sgfHXlXUriHSH4feMUvpRSkhnD1roVSEvz86t29nOBiQidd6odeHLcp4QGJJiRHWJFYqkxW53NICS1DQCBwnhc7KA5+2RktNqOePGRFyNMKlkx67di0qNm5EFieCY0AmU08IvaazTtI5HpCQFZHLKStWoFb4MIOiOXbNxwddotlEQFAzippTrmaYIm2IipImz3muoYGnYmNApkYP6axTsBEoFt29T6Kzdv48qUzNt2fmzMLtwgD86e0qvHyuAEebqhEZNDqcRA0IvZ0iS3G4tlZKTr1iyRKuJNNAZuFruhtG4jc+M4JNjy/gUargQlfvwaKkRwYiJswPh70X4vOuGFw7pkdc3PKRa9AkndRcOr5rFyzJydhuscB0MQ8lwm+hdYkREdhRUMDWYpppikNRpteXsXB4HAlN0UYzUTXrrPOmOwWDelOHj6mvKkaZyReLF2us5DxP+rSP1VKGosiT5lBPfWpq6qj9KHyEpkTIy8sbCVGZcYCM+uXQWaeI7hYaFOoXuiB0cRiGczrrtAoHh/P/yiEGbl2HAZnSWC1HcVr2QlFI8qQ5169fx5UrV6SwE3kbhY98/vnnGBwcHM8pEh7wB8aATGmslrOx6upQFOWkOQRPY2Oj1BMvJ4Kj8BGK4aJJOF1NpPNAA+KWeWVAOBSFLQgDwmJAGBDWuOv/ATStANXfGCYwAAAAAElFTkSuQmCCUEsHCPTtjTDxEwAA7BMAAFBLAwQUAAgACABIVFdDAAAAAAAAAAAAAAAAEgAAAGdlb2dlYnJhX21hY3JvLnhtbNVXTVPbMBA9l1+h0d12/JUPJoYhYQLMQJkpHHpVbCVRsSXXUgjh13cl2ZA0EJqWMCUHW16v3q7ek9ab/vFDkaN7WkkmeIJ9t4UR5anIGJ8meK4mThcfHx30p1RM6bgiaCKqgqgER9oTPUh2yMVXUlBZkpTepDNakEuREmXgZkqVh563WCzcBsAV1dSbTpX7IDOMIDiXCa4HhwC3NmkRGveg1fK971eXFt5hXCrCU4oRJFaQtBIoLTKdRIJTylUlGEArIfKXbOc0L3VElgo+YjnVYzkTiwt+C28HpEqwquYAnopyOSSlXomsbU28C17OFSKtBJ9gRPwED+AWJHiIvcbleq4anzNjhXASQFKNhxRTNjKZq5mo9CgjSlvAk+a0gJSRWpZgKQXjCqOcjGmu4x0dfOnrfJEY/6CparKt35sHT/vA66HIRYUAHaSamus4wUEcQ9i8nBGwgIjGOSdLWqF7kmuv2gJ4VyKja1bCWWG0RVJRoNEH7kpKMzOy+cKgBDizfyYkl3U6qRBVJtGD3jmdCKMlZOKC76PddMbHLPWGPdYxw1WrWuarqfS9mqU3+Bp8KF/Xk4mkSi/SCQKzRr+9bzodc2YhlBO7wd75HH7+/ef7brchrN19F8ZSURSEZ4ibenPFMkuansueSsXAlophTc9zfTi1cDXIGwKcbgpgV7ldgXZkJNC3sb3tvKVje2h72/R5Zjl2o4bkKOyt/uKPo/zkNcpHu1A++mDKt7DadYOOpdV3O901Wvexk7+R5cuMnm4wmm1ntAKkhq7sLT5Xju1rVaT1Hmx23Hh9axpqW2673Vr92a+UE3ZcP4ibes44tTyqGUvvOJXQIwRNDWrZwTnLMqq7KjuH/uR2itTXBLOizFnK1BOXua5aF1xBN0ZNnyBt4it83FFa3gL0Nb+tCJe6G7M+zR7cRcy6Io02xKR/Lib9X8R0QjfyjYA9c0hAsghajV70aQV7Av5NtszKRjdkO9ulqp391Zd8H0Ut7IRux++GsW2ZnAC+0H7UjdrN4QyNmh3oF6N29E/lzVvtwvWz6dX1oPl/cvQLUEsHCKaxR5IdAwAAEg0AAFBLAwQUAAgACABIVFdDAAAAAAAAAAAAAAAAFgAAAGdlb2dlYnJhX2phdmFzY3JpcHQuanNLK81LLsnMz1NIT0/yz/PMyyzR0FSoruUCAFBLBwhFzN5dGgAAABgAAABQSwMEFAAIAAgASFRXQwAAAAAAAAAAAAAAAAwAAABnZW9nZWJyYS54bWzlnHlv20YWwP9uP8VAKBYJNqLnHrK1Wzguig2QNI6dXRTbbQuKomTGEqmQlI9s97vvm4MSqSuiLcdKY8ShyBnO8X7vmhHpwx9uxiN0FedFkqVHHeLhDorTKOsn6fCoMy0HXb/zw/dfHw7jbBj38hANsnwclkcdrmsm/aOOH8YDxjHvyp4YdLmKBt0QR7Ib9Eg/GPAIDyjtIHRTJN+m2c/hOC4mYRSfRxfxOHyZRWFpOr4oy8m3BwfX19de1ZWX5cOD4bDn3RT9DoJhpsVRx334Fppr3HTNTHWKMTn45dVL23w3SYsyTKO4g/QUpsn3X391eJ2k/ewaXSf98gImTH2Yx0WcDC9gUkrAUA90rQlIZBJHZXIVF3Bv7dRMuhxPOqZamOryr+wnNJrNp4P6yVXSj/OjDvag0SxP4rR0hcR1clDdfniVxNe2Hf3JdME7qMyyUS/UTaA//0QUU4ye6QOxBwoHKW0RttcwswdqD9wehK3D7e3cVuW2Drd1OOugq6RIeqP4qDMIRwXILEkHOfCanRfl7Sg243EX5tMlz2BORfIBKjMMArVChusYP9O/En65LjhoTpLUei3zactOqy4JVf72fdJ7zZRVnVIhl/ukYs085YZO7cS3mqioyRa6Mv/M71KPbNM0F3u05/frUPJPMsXDg8pWDp15oOJC13XqU8bjQhsMC5AItN4TJMA4pAI1F4gEcFAUgTkgIhAXcEp8JPVRIaaggCOGfKTrEYaMdQgf/uPKNCaRgMb0VQVGiQh0xJFgiBij4ghMCRnDBCOlDGoIgQTcpLsnVDfBJOISzpiPOIxR26QiUJHBjXAO3VPECGL6ZqIQlUjq9gjXti59PXRokiKJkSS6QTBrMGlrzlDfR0zPRprRwU/lbJJ0Mi0bsorG/epjmU1mUKA2eKa5w7OequEPvzochb14BDHiXCNF6CocadMwHQ2ytEQzy7TXhnk4uUii4jwuS7irQO/Cq/BlWMY3P0Htourb1I2ytDjNs/IkG03HaYFQlI3wbMzZiNQ+09mo4YTVCni9QNQKZO2zWtlvBiVoWsTQf5YXVfWw33+ha8x9BEjydTq6fZ7H4eUkS5rTODww4eYwnkajpJ+E6b9Aa3UvWi5oHn2046qij/D9aiRZ3j+/LUCX0c2/4zyDmkp4gnAsGYWPPAB/cmtLmCQeF0LhQDEfA3IoKqJQWyGjHg04FxJTwTlRcM/qEtdvfDXjE97E86kOc23gtZMXxfNsNL9kZn8STsppbtIG8JG5ntJxOhzFRkOMgUNMji572c25VQ1m23p7O4EzbEfQGxqpIx3zoNj835uV6kFtKtfNzcpJQE0Nc+zZo6kFamsH5SZJqgmSWTdJYVwa7jh7qdyVVnod26dpUr6sTsokupxPUt/w83Tci2eq02yT7KrNw4MF3Tq8jPM0HjlVBozTbFpYy6xpeT+OkjGc2gInklCD+icMwF7tx8M8rgY+MsmYFZgpxXUlXbpsmvopz8Yv0qu3oAULAzg8qEZ5WER5MtHahnoQBy7juT71kyKEMNKv36dtD6Ye6XAB4im1aMAqp+VFlpt0C5wJHLXJjeIx5FqoNIqVTsdxnkQzQfdN3gaDmrpxK0/YkWspo6z3DrzcLCLae+YYoXimYFQIq2CCWQXTx3A0uQh1zkecsoW3cd4QkGnzVdZ33bt6xUgni2icQCwEOx2HN0YfUdgrwA2WkC8DkXSeL9vxVW4EY52Nwy1c6A+3MLhAfxgkN/HMx4Ogkg+gGGFjR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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
 * Estimated Time: 90 minutes
 * Materials/ Resources needed:
 * 1) Laptop, geogebra file, projector and a pointer.
 * 2) Students' individual construction materials.
 * Prerequisites/Instructions, if any
 * 1) The students should have prior knowledge of a circle, tangent and the limiting case of a secant as a tangent.
 * 2) They should understand that a tangent is always perpendicular to the radius of the circle.
 * 3) They should know construction of a tangent to a given point.
 * 4) If the same straight line is a tangent to two or more circles, then it is called a common tangent.
 * 5) If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent.
 * 6) Note: In general,
 * The two circles are named as C1 and C2
 * The distance between the centre of two circles is 'd'
 * Radius of one circle is taken as 'R' and other as 'r'
 * The length of tangent is 't'
 * Multimedia resources:Laptop
 * Website interactives/ links/ / Geogebra Applets : This geogebra file was created by Mallikarjun sudi of Yadgir.

The teacher can explain the step by step construction of Direct common tangent and with an example. Developmental Questions:
 * Process:
 * 1) What is a tangent
 * 2) What is a common tangent ?
 * 3) What is a direct common tangent ?
 * 4) What is R and r  ?
 * 5) What does the length OA represent here ?
 * 6) Why was a third circle constructed ?
 * 7) Let us try to construct direct common tangent without the third circle and see.
 * 8) What should be the radius of the third circle ?
 * 9) Why was OA bisected and semi circle constructed ?
 * 10) What were OB and OC extended ?
 * 11) What can you say about lines AB and AC ?
 * 12) Name the direct common tangents.
 * 13) At what points is the tangent touching the circles ?
 * 14) Identify the two right angled triangles formed from the figure ? What do you understand ?
 * Evaluation:
 * 1) Is the student able to comprehend the sequence of steps in constructing the tangent.
 * 2) Is the student able to identify error areas while constructing ?
 * 3) Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
 * 4) Is the student able to appreciate that the direct common tangents from the same external point are equal and subtend equal angles at the center.
 * Question Corner:
 * 1) What do you think are the applications of tangent constructions ?
 * 2) What is the formula to find the length of direct common tangent ?
 * 3) Can a direct common tangent be drawn to two circles one inside the other ?
 * 4) Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
 * 5) What are properties of direct common tangents ?
 * 6) [Note for  teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.] Examine with the help of following geogebra file made by Ranjani.

<ggb_applet width="1280" height="600" version="4.0" ggbBase64="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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />

Activity No # 2. Construction of Transverse common tangent
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Developmental Questions
 * Estimated Time: 45 minutes
 * Materials/ Resources needed:
 * 1) Laptop, geogebra file, projector and a pointer.
 * 2) Students' individual construction materials.
 * Prerequisites/Instructions, if any
 * 1) The students should have prior knowledge of a circle, tangent and direct and transverse common tangents.
 * 2) They should understand that a tangent is always perpendicular to the radius of the circle.
 * 3) They should know construction of a tangent to a given point.
 * 4) If the same straight line is a tangent to two or more circles, then it is called a common tangent.
 * 5) If the centres of the circles lie on opposite side of the common tangent, then the tangent is called a transverse common tangent.
 * 6) Note: In general,
 * The two circles are named as C1 and C2
 * The distance between the centre of two circles is 'd'
 * Radius of one circle is taken as 'R' and other as 'r'
 * The length of tangent is 't'
 * Multimedia resources: Laptop
 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 1) The teacher can explain the step by step construction of Transverse common tangent.
 * 1) What is a transverse common tangent ?
 * 2) What is the radius of the third circle ?
 * 3) What is the difference in finding the radius of the third circle in constructing Dct and that of Tct ?
 * 4) Why was a third circle constructed ?
 * 5) Let us try to construct transverse common tangent without the third circle and see.
 * 6) Name the transverse common tangents.
 * 7) At what points is the tangent touching the circles ?
 * Evaluation:
 * 1) Is the student able to comprehend the sequence of steps in constructing the tangent.
 * 2) Is the student able to identify error areas while constructing ?
 * 3) Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
 * 4) Is the student able to understand the difference in the construction protocol between direct common tangent and transverse common tangent ?
 * Question Corner:# What do you think are the applications of tangent constructions ?
 * 1) What is the formula to find the length of transverse common tangent ?
 * 2) Can a direct common tangent be drawn to two circles one inside the other ?
 * 3) What are properties of transverse common tangents ?
 * Evaluation:
 * 1) Were the students able to comprehend the steps in transverse common tangent construction ?
 * Question Corner:
 * 1) Can you construct a transverse common tangent without the third circle ?

Learning objectives

 * 1) A quadrilateral ABCD is called cyclic if all four vertices of it 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) Circles and quadrilaterals should have been covered.
 * Multimedia resources : Laptop
 * Website interactives/ links/ / Geogebra Applets; This geogebra file was created by ITfC-Edu-Team.
 * Process:
 * 1) Show the geogebra file.
 * 2) 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) Name this special quadrilateral.

Activity No # 2.Properties of a Cyclic quadrilateral
coloured paper, pair if 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 Note: Refer the above geogebra file to understand the below mentioned labelling.,br> Developmental Questions: Name the two properties of cyclic quarilaterals.
 * Estimated Time: 45 minutes
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * 1) Circles and quadrilaterals should have been covered.
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 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) Make a replica of cyclic quadrilateral ABCD using carbon paper.
 * 6) Cut the replica into 4 parts such that each part contains one angle.
 * 7) Draw a straight line on a paper.
 * 8) Place angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
 * 9) Place angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
 * 10) Produce AB to form a ray AE such that exterior angle CBE is formed.
 * 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:

= Hints for difficult problems = Please click here for solution.
 * 1) Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ

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