Difference between revisions of "The longest chord passes through the centre of the circle"
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===Notes for teachers=== | ===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 many ways. The theorems that involve chords of a circle are : | + | # A chord is a straight line joining 2 points on the circumference of a circle. |
− | Perpendicular bisector of a chord passes through the center of a circle. | + | # Chords within a circle can be related in many ways. |
− | Congruent chords are equidistant from the center of a circle. | + | # The theorems that involve chords of a circle are : |
− | If two chords in a circle are congruent, then their intercepted arcs are congruent. | + | * Perpendicular bisector of a chord passes through the center of a circle. |
− | If two chords in a circle are congruent, then they determine two central angles that are congruent. | + | * 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.] === | ===Activity No 1[Theorem 1: Perpendicular bisector of a chord passes through the center of a circle.] === |
Revision as of 10:03, 3 December 2013
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
- 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.
- Then she can test if all students know 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
Learning objectives
Notes for teachers
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 #3.TANGENT
Learning objectives
Notes for teachers
Activity No # 1. Construction of tangent from an external point .
- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner
Activity No # Construction of Direct common tangent
- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner
Activity No # Construction of Transverse common tangent
- 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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