Difference between revisions of "The longest chord passes through the centre of the circle"
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==Concept # Cyclic quadrilateral== | ==Concept # Cyclic quadrilateral== |
Revision as of 20:59, 9 July 2014
Philosophy of Mathematics |
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Concept Map
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Textbook
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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
- Meaning of circle and chord.
- Method to measure the perpendicular distance of the chord from the centre of the circle.
- Properties of chord.
- Able to relate chord properties to find unknown measures in a circle.
- Apply chord properties for proof of further theorems in circles.
- 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.
Activities
- Activity No 1 - Theorem 1: Perpendicular bisector of a chord passes through the center of a circle
- Activity No 2 - Theorem 2.Congruent chords are equidistant from the center of a circle
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
Activities
- Activity #1 - Understanding secant and tangent using Geogebra
Concept #3 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
Activities
- Activity #1 - Construction of Direct common tangent
- Activity #2 - Construction of Transverse common tangent
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.
- Process:
- Show the geogebra file.
- 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
- Circles and quadrilaterals should have been covered.
- 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.,br>
- 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.
Hints for difficult problems
- Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ
Please click here for solution.
Project Ideas
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