The longest chord passes through the centre of the circle
Philosophy of Mathematics |
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Additional Information
Useful websites
- www.regentsprep.com conatins good objective problems on chords and secants
- www.mathwarehouse.com contains good content on circles for different classes
- staff.argyll contains good simulations
Reference Books
= Teaching Outlines Chord and its related theorems
Concept #1 CHORD
Learning objectives
The students should be able to:
- Recall the meaning of circle and chord.
- They should know the method to measure the perpendicular distance of the chord from the centre of the circle.
- State Properties of chord.
- By studying the theorems related to chords, the students should know that a chord in a circle is an important concept .
- They should be able to relate chord properties to find unknown measures in a circle.
- They should be able to apply chord properties for proof of further theorems in circles.
- The students should understand the meaning of congruent chords.
Notes for teachers
- A chord is a straight line joining 2 points on the circumference of a circle.
- Chords within a circle can be related in many ways.
- 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.]
- Estimated Time
20 minutes
- Materials/ Resources needed:
Laptop, Geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any
- The students should know the basic concepts of a circle and its related terms.
- They should have prior knowledge of chord and construction of perpendicular bisector to the chord.
- Multimedia resources: Laptop
- Website interactives/ links/ / Geogebra Applets
- Process:
- Show the children the geogebra file.
- Let them identify the chord. Ask them to define a chord.
- Let them recall what a perpendicular bisector is.
- Show them the second chord.
- Let students observe if everytime the perpendicular bisector of the chord passes through the centre of the circle.
- Developmental Questions:
- What is a chord ?
- At how many points on the circumference does the chord touch a circle .
- What is a bisector ?
- What is a perpendicular bisector ?
- In each case the perpendicular bisector passes through which point ?
- Can anyone explain why does the perpendicular bisector always passes through the centre of the circle ?
- Evaluation
- What is the angle formed at the point of intersection of chord and radius ?
- Are the students able to understand what a perpendicular bisector is ?
- 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:
- What do you infer ?
- 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.]
- Estimated Time :40 minutes.
- Materials/ Resources needed:
Laptop, geogebra,projector and a pointer.
- Prerequisites/Instructions, if any
- The students should have prior knowledge of a circle, its centre, radius, circumference and a chord.
- They should know that the length of the chord means its perpendicular distance from the centre.
- They should know to draw perpendicular bisector to a given chord.
- They should know the meaning of the term congruent and equidistant.
- Multimedia resources: Laptop, geogebra file, projector and a pointer.
- Website interactives/ links/ / Geogebra Applets
- Process:
- The teacher can reiterate the prior knowledge on circles.
- Revise the procedure of drawing chords of given length accurately in a circle.
- Revise what congruent chords mean.
- Show geogebra file and explain to help them understand the theorem.
- Developmental Questions:
- What is a chord ?
- Name the centre of the circle.
- How do you draw congruent chords in a circle ?
- How many chords do you see in the figure ? Name them.
- If both the chords are congruent, what can you say about the length of both the chords ?
- How can we measure the length of the chord ?
- What is the procedure to draw perpendicular bisector ?
- What does theorem 1 say ? Do you all remember ?
- What is the length of both chords here ?
- What can you conclude ?
- Repeat this for circles of different radii and for different lengths of congruent chords.
- Evaluation:
- Were the students able to comprehend the drawing of congruent chords in a circle ?
- Were the students able to comprehend why congruent chords are always equal for a given circle. Let any student explain the analogy.
- Are the students able to understand that this theorem can be very useful in solving problems related to circles and triangles ?
- Question Corner:
- What is a chord ?
- What are congruent chords ?
- 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
Concept #2.Secant and Tangent
Learning objectives
- The secant is a line passing through a circle touching it at any two points on the circumference.
- A tangent is a line toucing the circle at only one point on the circumference.
Notes for teachers
Activity No #
- Estimated Time: 15 minutes
- Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any:
- The students should have a prior knowledge about a circle and its basic parts and terms.
- They should know the clear distinction between radius, diameter, chord, secant and tangent.
- Multimedia resources : Laptop and projector
- Website interactives/ links/ / Geogebra Applets
- Process:
- The teacher can show the geogebra file.
- Move the points on circumference and explain secant.
- When both endpoints of secant meet, it becomes a tangent.
Developmental Questions:
- Name the points on the circumference of the circle.
- At how many points is the line touching the circle ?
- What is the line called ?
- Evaluation
- What is the difference between the secant and a tangent?
- What is the difference between the chord and a secant ?
- Question Corner
- Can you draw a secant touching 3 points on the circle ?
- At how many points does a tangent touch a circle ?
- How many tangents can be drawn to a circle ?
- How many tangents can be drawn to a circle at any one given point ?
- 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
Concept # Construction of tangents
Learning objectives
- The students should know that tangent is a straight line touching the circle at one and only point.
- They should understand that a tangent is perpendicular to the radius of the circle.
- The construction protocol of a tangent.
- Constructing a tangent to a point on the circle.
- Constructing tangents to a circle from external point at a given distance.
- A tangent that is common to two circles is called a common tangent.
- A common tangent with both centres on the same side of the tangent is called a direct common tangent.
- A common tangent with both centres on either side of the tangent is called a transverse common tangent.
Notes for teachers
Activity No # Construction of Direct common tangent
- Estimated Time: 90 minutes
- Materials/ Resources needed:
- Laptop, geogebra file, projector and a pointer.
- Students' individual construction materials.
- Prerequisites/Instructions, if any
- The students should have prior knowledge of a circle , tangent and the limiting case of a secant as a tangent.
- They should understand that a tangent is always perpendicular to the radius of the circle.
- They should know construction of a tangent to a given point.
- If the same straight line is a tangent to two or more circles, then it is called a common tangent.
- If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct 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
- Process:
The teacher can explain the step by step construction of Direct common tangent and with an example. [Note for teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.] Developmental Questions:
- What is a tangent
- What is a common tangent ?
- What is a direct common tangent ?
- What is R and r ?
- What does the length OA represent here ?
- Why was a third circle constructed ?
- Let us try to construct direct common tangent without the third circle and see.
- What should be the radius of the third circle ?
- Why was OA bisected and semi circle constructed ?
- What were OB and OC extended ?
- What can you say about lines AB and AC ?
- Name the direct common tangents .
- At what points is the tangent touching the circles ?
- Identify the two right angled triangles formed from the figure ? What do you understand ?
- 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 appreciate that the direct common tangents from the same external point are equal and subtend equal angles at the center.
- Question Corner:
- What do you think are the applications of tangent constructions ?
- What is the formula to find the length of direct common tangent ?
- Can a direct common tangent be drawn to two circles one inside the other ?
- Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
- What are properties of direct common tangents ?
Activity No # Construction of Transverse common tangent
- Estimated Time: 45 minutes
- Materials/ Resources needed:
- Laptop, geogebra file, projector and a pointer.
- Students' individual construction materials.
- Prerequisites/Instructions, if any
- The students should have prior knowledge of a circle , tangent and direct and transverse common tangents .
- They should understand that a tangent is always perpendicular to the radius of the circle.
- They should know construction of a tangent to a given point.
- If the same straight line is a tangent to two or more circles, then it is called a common tangent.
- 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
- 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
- The students should learn that a quadrilateral ABCD is called cyclic if all four vertices of it lie on a circle.
- They should know that 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#1Cyclic quadrilateral
- Estimated Time 10 minutes
- Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any
- The students should know a circle and its parts.
- They should know that a quadrilateral is a 4 sided closed figure.
- Multimedia resources : Laptop
- Website interactives/ links/ / Geogebra Applets
- Process:
- The teacher can recall the concept of a circle, quadrilateral, circumcircle.
- 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.
Developmental Questions:
- What two figures do you see in the figure ?
- Name the vertices of the quadrilateral.
- Where are all the 4 vertices situated ?
- Name the opposite interior angles of the quadrilateral.
- What do you observe about them.
- Evaluation:
- Compare the cyclic quadrilateral to circumcircle.
- Question Corner
- Name this special quadrilateral.
Activity No # 2.Properties of a Cyclic quadrilateral
- Estimated Time: 45 minutes
- Materials/ Resources needed
coloured paper, pair if 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:
Note: Refer the above geogebra file to understand the below mentioned labelling.
- 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.
- 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 angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
- Place angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
- Produce AB to form a ray AE such that exterior angle CBE is formed.
- 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.
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