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| + | Investigating the diameter is the longest chord of a circle. |
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| − | [http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story 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 = | + | ===Objectives=== |
| − | __FORCETOC__
| + | To understand longest chord passes through the centre and it is the diameter |
| − | <mm>[[circles_and_lines.mm|flash]]</mm>
| + | ===Estimated Time=== |
| | + | 30 minutes |
| | | | |
| − | = Textbook = | + | ===Prerequisites/Instructions, prior preparations, if any=== |
| − | To add textbook links, please follow these instructions to:
| + | Prior knowledge of point, lines, angles, polygons |
| − | ([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])
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| | | | |
| − | =Additional Information= | + | ===Materials/ Resources needed=== |
| − | ==Useful websites== | + | * Digital : Computer, geogebra application, projector. |
| − | #[http://www.regentsprep.org/Regents/math/geometry/GP14/PracCircleSegments.htm www.regentsprep.com] conatins good objective problems on chords and secants <br>
| + | * Non digital : Worksheet and pencil, compass, strings |
| − | #[http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php www.mathwarehouse.com] contains good content on circles for different classes<br>
| + | * Geogebra files : [https://ggbm.at/c4eg7q2u Diameter is longest chord.ggb] |
| − | #[http://staff.argyll.epsb.ca/jreed/math20p/circles/tangent.htm staff.argyll] contains good simulations
| + | {{Geogebra|c4eg7q2u}} |
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| − | ==Reference Books== | + | ===Process (How to do the activity)=== |
| | + | Use the geogebra file to show how diameter is the longest chord. |
| | | | |
| − | = Teaching Outlines =
| + | Move the points on the circle to show the changes in the triangle. |
| − | Chord and its related theorems
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| − | ==Concept #1 Chord==
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| − | ===Learning objectives===
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| − | # Meaning of circle and chord.
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| − | # Method to measure the perpendicular distance of the chord from the centre of the circle.
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| − | # Properties of chord.
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| − | # Able to relate chord properties to find unknown measures in a circle.
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| − | # Apply chord properties for proof of further theorems in circles.
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| − | # Understand the meaning of congruent chords.
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| − | ===Notes for teachers===
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| − | # A chord is a straight line joining 2 points on the circumference of a circle.
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| − | # Chords within a circle can be related in many ways.
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| − | # The theorems that involve chords of a circle are :
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| − | * Perpendicular bisector of a chord passes through the center of a circle.
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| − | * Congruent chords are equidistant from the center of a circle.
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| − | * If two chords in a circle are congruent, then their intercepted arcs are congruent.
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| − | * If two chords in a circle are congruent, then they determine two central angles that are congruent.
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| | | | |
| − | ===Activities===
| + | What is the condition with respect to sides for formation of a triangle. Sum of two sides is larger than the third side. |
| − | #Activity No 1 - [[Circles_and_lines_activity_1|Theorem 1: Perpendicular bisector of a chord passes through the center of a circle]]
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| − | #Activity No 2 - [[Circles_and_lines_activity_2|Theorem 2.Congruent chords are equidistant from the center of a circle]]
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| | | | |
| − | ===Activity No # ===
| + | Compare the chord length with sum of two radii. When is the triangle reduced to a line segment. |
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| − | |<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;">
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| − | ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | *Estimated Time
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| − | *Materials/ Resources needed
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| − | *Prerequisites/Instructions, if any
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| − | *Multimedia resources
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| − | *Website interactives/ links/ / Geogebra Applets
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| − | *Process/ Developmental Questions
| + | What can you conclude about the chord? When is it the largest? |
| − | *Evaluation
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| − | *Question Corner
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| | | | |
| − | ===Activity No # ===
| + | [[Category:Circles]] |
| − | {| style="height:10px; float:right; align:center;"
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| − | |<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;">
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| − | ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | *Estimated Time
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| − | *Materials/ Resources needed
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| − | *Prerequisites/Instructions, if any
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| − | *Multimedia resources
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| − | *Website interactives/ links/ / Geogebra Applets
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| − | *Process/ Developmental Questions
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| − | *Evaluation
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| − | *Question Corner
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| − | | |
| − | ==Concept #2.Secant and Tangent==
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| − | ===Learning objectives===
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| − | # The secant is a line passing through a circle touching it at any two points on the circumference.
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| − | # A tangent is a line toucing the circle at only one point on the circumference.
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| − | ===Notes for teachers===
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| − | ===Activity No # 1.Understanding Secant and Tangent using geogebra. ===
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| − | {| style="height:10px; float:right; align:center;"
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| − | |<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;">
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| − | ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | *Estimated Time: 15 minutes
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| − | *Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
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| − | *Prerequisites/Instructions, if any:
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| − | # The students should have a prior knowledge about a circle and its basic parts and terms.
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| − | # They should know the clear distinction between radius, diameter, chord, secant and tangent.
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| − | *Multimedia resources : Laptop and projector
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| − | *Website interactives/ links/ / Geogebra Applets
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| − | This geogebra file has been made by ITfC-Edu-team
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| − | *Process:
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| − | # The teacher can show the geogebra file.
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| − | # Move the points on circumference and explain secant.
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| − | # When both endpoints of secant meet, it becomes a tangent.
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| − | Developmental Questions:
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| − | # Name the points on the circumference of the circle.
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| − | # At how many points is the line touching the circle ?
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| − | # What is the line called ?
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| − | *Evaluation
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| − | # What is the difference between the secant and a tangent?
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| − | # What is the difference between the chord and a secant ?
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| − | *Question Corner
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| − | # Can you draw a secant touching 3 points on the circle ?
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| − | # At how many points does a tangent touch a circle ?
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| − | # How many tangents can be drawn to a circle ?
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| − | # How many tangents can be drawn to a circle at any one given point ?
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| − | # How many parallel tangents can a circle have at the most ?
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| − | | |
| − | ===Activity No # ===
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| − | {| style="height:10px; float:right; align:center;"
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| − | |<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;">
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| − | ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | *Estimated Time
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| − | *Materials/ Resources needed
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| − | *Prerequisites/Instructions, if any
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| − | *Multimedia resources
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| − | *Website interactives/ links/ / Geogebra Applets
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| − | *Process/ Developmental Questions
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| − | *Evaluation
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| − | *Question Corner
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| − | | |
| − | ==Concept # Construction of tangents==
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| − | ===Learning objectives===
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| − | # The students should know that tangent is a straight line touching the circle at one and only point.
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| − | # They should understand that a tangent is perpendicular to the radius of the circle.
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| − | # The construction protocol of a tangent.
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| − | # Constructing a tangent to a point on the circle.
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| − | # Constructing tangents to a circle from external point at a given distance.
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| − | # A tangent that is common to two circles is called a common tangent.
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| − | # A common tangent with both centres on the same side of the tangent is called a direct common tangent.
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| − | # A common tangent with both centres on either side of the tangent is called a transverse common tangent.
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| − | | |
| − | ===Notes for teachers===
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| − | ===Activity No # 1. Construction of Direct common tangent ===
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| − | {| style="height:10px; float:right; align:center;"
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| − | |<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;">
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| − | ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | *Estimated Time: 90 minutes
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| − | *Materials/ Resources needed:
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| − | # Laptop, geogebra file, projector and a pointer.
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| − | # Students' individual construction materials.
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| − | *Prerequisites/Instructions, if any
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| − | # The students should have prior knowledge of a circle , tangent and the limiting case of a secant as a tangent.
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| − | # They should understand that a tangent is always perpendicular to the radius of the circle.
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| − | # They should know construction of a tangent to a given point.
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| − | # If the same straight line is a tangent to two or more circles, then it is called a common tangent.
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| − | # If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent.
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| − | # Note: In general,
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| − | *The two circles are named as C1 and C2
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| − | * The distance between the centre of two circles is 'd'
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| − | * Radius of one circle is taken as 'R' and other as 'r'
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| − | * The length of tangent is 't'
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| − | *Multimedia resources:Laptop
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| − | *Website interactives/ links/ / Geogebra Applets : This geogebra file was created by Mallikarjun sudi of Yadgir.
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| − | <ggb_applet width="1280" height="600" version="4.0" 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| − | *Process:
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| − | The teacher can explain the step by step construction of Direct common tangent and with an example.<br>
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| − | Developmental Questions:
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| − | # What is a tangent
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| − | # What is a common tangent ?
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| − | # What is a direct common tangent ?
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| − | # What is R and r ?
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| − | # What does the length OA represent here ?
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| − | # Why was a third circle constructed ?
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| − | # Let us try to construct direct common tangent without the third circle and see.
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| − | # What should be the radius of the third circle ?
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| − | # Why was OA bisected and semi circle constructed ?
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| − | # What were OB and OC extended ?
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| − | # What can you say about lines AB and AC ?
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| − | # Name the direct common tangents .
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| − | # At what points is the tangent touching the circles ?
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| − | # Identify the two right angled triangles formed from the figure ? What do you understand ?
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| − | *Evaluation:
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| − | # Is the student able to comprehend the sequence of steps in constructing the tangent.
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| − | # Is the student able to identify error areas while constructing ?
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| − | # Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
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| − | # 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.
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| − | *Question Corner:
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| − | # What do you think are the applications of tangent constructions ?
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| − | # What is the formula to find the length of direct common tangent ?
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| − | # Can a direct common tangent be drawn to two circles one inside the other ?
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| − | # Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
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| − | # What are properties of direct common tangents ?
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| − | # [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.
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| − | ===Activity No # 2. Construction of Transverse common tangent===
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| − | {| style="height:10px; float:right; align:center;"
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| − | |<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;">
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| − | ''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | *Estimated Time: 45 minutes
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| − | *Materials/ Resources needed:
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| − | # Laptop, geogebra file, projector and a pointer.
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| − | # Students' individual construction materials.
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| − | *Prerequisites/Instructions, if any
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| − | # The students should have prior knowledge of a circle , tangent and direct and transverse common tangents .
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| − | # They should understand that a tangent is always perpendicular to the radius of the circle.
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| − | # They should know construction of a tangent to a given point.
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| − | # If the same straight line is a tangent to two or more circles, then it is called a common tangent.
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| − | # If the centres of the circles lie on opposite side of the common tangent, then the tangent is called a transverse common tangent.
| |
| − | # 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
| |
| − | <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 explain the step by step construction of Transverse common tangent.
| |
| − | Developmental Questions
| |
| − | # What is a transverse common tangent ?
| |
| − | # What is the radius of the third circle ?
| |
| − | # What is the difference in finding the radius of the third circle in constructing Dct and that of Tct ?
| |
| − | # Why was a third circle constructed ?
| |
| − | # Let us try to construct transverse common tangent without the third circle and see.
| |
| − | # Name the transverse common tangents .
| |
| − | # At what points is the tangent touching the circles ?
| |
| − | *Evaluation:
| |
| − | # Is the student able to comprehend the sequence of steps in constructing the tangent.
| |
| − | # Is the student able to identify error areas while constructing ?
| |
| − | # Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
| |
| − | # 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 ?
| |
| − | # What is the formula to find the length of transverse common tangent ?
| |
| − | # Can a direct common tangent be drawn to two circles one inside the other ?
| |
| − | # What are properties of transverse common tangents ?
| |
| − | *Evaluation:
| |
| − | # Were the students able to comprehend the steps in transverse common tangent construction ?
| |
| − | *Question Corner:
| |
| − | # Can you construct a transverse common tangent without the third circle ?
| |
| − | | |
| − | ==Concept # Cyclic quadrilateral==
| |
| − | ===Learning objectives===
| |
| − | # 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
| |
| − | ===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
| |
| − | # Circles and quadrilaterals should have been covered.
| |
| − | *Multimedia resources : Laptop
| |
| − | *Website interactives/ links/ / Geogebra Applets; This geogebra file was created by ITfC-Edu-Team.
| |
| − | <ggb_applet width="1282" height="601" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
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| − | *Process:
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| − | # Show the geogebra file.
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| − | # Move points, the vertices of the quadrilateral and let the students observe the sum of opposite interior angles.
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| − | Developmental Questions:
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| − | # What two figures do you see in the figure ?
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| − | # Name the vertices of the quadrilateral.
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| − | # Where are all the 4 vertices situated ?
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| − | # Name the opposite interior angles of the quadrilateral.
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| − | # What do you observe about them.
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| − | *Evaluation:
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| − | # Compare the cyclic quadrilateral to circumcircle.
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| − | *Question Corner
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| − | # Name this special quadrilateral.
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| − | | |
| − | ===Activity No # 2.Properties of a Cyclic quadrilateral===
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| − | *Estimated Time: 45 minutes
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| − | *Materials/ Resources needed
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| − | coloured paper, pair if scissors, sketch pen, carbon paper, geometry box
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| − | *Prerequisites/Instructions, if any
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| − | # Circles and quadrilaterals should have been covered.
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| − | *Multimedia resources
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| − | *Website interactives/ links/ / Geogebra Applets
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| − | This activity has been taken from the website http://mykhmsmathclass.blogspot.in/2007/11/class-ix-activity-16.html
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| − | *Process:
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| − | Note: Refer the above geogebra file to understand the below mentioned labelling.,br>
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| − | # Draw a circle of any radius on a coloured paper and cut it.
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| − | # Paste the circle cut out on a rectangular sheet of paper.
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| − | # By paper folding get chords AB, BC, CD and DA in order.
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| − | # Draw AB, BC, CD & DA. A cyclic quadrilateral ABCD is obtained.
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| − | # Make a replica of cyclic quadrilateral ABCD using carbon paper.
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| − | # Cut the replica into 4 parts such that each part contains one angle .
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| − | # Draw a straight line on a paper.
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| − | # Place angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
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| − | # Place angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
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| − | # Produce AB to form a ray AE such that exterior angle CBE is formed.
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| − | # Make a replica of angle ADC and place it on angle CBE . Write the observation.
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| − | Developmental Questions:
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| − | # How do you take radius ?
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| − | # What is the circumference ?
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| − | # What is a chord ?
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| − | # What is a quadrilateral ?
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| − | # Where are all four vertices of a quadrilateral located ?
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| − | # What part are we trying to cut and compare ?
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| − | # What can you infer ?
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| − | *Evaluation:
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| − | # Angle BAD and angle BCD, when placed adjacent to each other on a straight line, completely cover the straight angle.What does this mean ?
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| − | # Angle ABC and angle ADC, when placed adjacent to each other on a straight line, completely cover the straight angle.What can you conclude ?
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| − | # Compare angle ADC with angle CBE.
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| − | *Question Corner:
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| − | Name the two properties of cyclic quarilaterals.
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| − | | |
| − | = Hints for difficult problems =
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| − | #Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ
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| − | Please click [http://karnatakaeducation.org.in/KOER/en/index.php/Class10_circles_tangents#Problem_1 here] for solution.
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| − | | |
| − | [[Class10_circles_tangents|here]]
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| − | | |
| − | = Project Ideas =
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| − | | |
| − | = Math Fun =
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| − | | |
| − | '''Usage'''
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| − | | |
| − | Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
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