cyclic quadrilateral

AIM:
By paper folding, cutting and pasting verify that
1) "The sum of interior opposite angles of cyclic quadrilateral is 180 degrees".
2) "In a cyclic quadrilateral the exterior angle is equal to interior opposite angle".

Material Required:
coloured paper, pair if scissors, sketch pen, carbon paper, geometry box

Procedure:
Step 1: Draw a circle of any radius on a coloured paper and cut it.
Step 2: Paste the circle cut out on a rectangular sheet of paper.
Step 3: By paper folding get chords AB, BC, CD and DA in order.
Step 4: Draw AB, BC, CD & DA. A cyclic quadrilateral ABCD is obtained.
Step 5: Make a replica of cyclic quadrilateral ABCD using carbon paper.
Step 6:Cut the replica into 4 parts such that each part contains one angle .
Step 7:Draw a straight line on a paper.
Step 8:Place angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
Step 9: Place angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
Step 10: Produce AB to form a ray AE such that exterior angle CBE is formed.
Step 11: Make a replica of angle ADC and place it on angle CBE . Write the observation.

Observations :

1) angle BAD and angle BCD , when placed adjacent to each other on a straight line, completely cover the straight angle. This means their sum is 180 degrees.

2) angle ABC and angle ADC , when placed adjacent to each other on a straight line, completely cover the straight angle. This means their sum is 180 degrees.


3) The replica of angle ADC completely covers angle CBE.

RESULT: a)The sum of either pair of opposite angle of a cyclic quadrilateral is 180 degrees.



b)In a cyclic quadrilateral, the exterior angle is equal to interior opposite angle.

Lengths of tangents from an external point

Aim By paper folding, cutting and pasting verify " Lengths of tangents drawn from an external point to a circle are equal".

Material Required Colored paper, scissors, geometry box, thread, glue

Procedure
Step 1 Draw a circle of any radius on a clolored paper an cut it.

Step 2 Take a rectangular sheet of paper and paste the circle cut out on it. Let O be the centre of the circle
Step 3 Mark a point P outside the circle.

Step 4 Fold the rectangular sheet in such a way that it passes through P and touches the circle at a point.

Step 5 On the crease, draw a tangent to the circle PA, where A is point of contact. Similarly get another tanget PB.

Step 6 Take a thread and paste its corner on P and cut the thread of length PA.

Step 7 Put the thread of length PA on PB.

Observations

The thread covering PA completely covers PB.

PA and PB are tangents of a circle.

Result

The lengths of tangents drawn from an external point to a circle are equal.

radius is perpendicular to tangent

Aim: By paper folding,cutting and pasting verify "radius of the circle is perpendicular to tangent through point of contact".

Material required: Coloured paper,pair of scissors,geometry box

Procedure:
1.Draw a circle of any radius and cut it.Mark the centre of circle as O.


2.Take a rectangular sheet and paste the cut -out on it.

3.Fold the rectangular sheet in such a way it just touches the circle.

Unfold it and on the crease draw a tangent PQ

4.Let A be the point of contact.

5.Form a crease joining OA and draw OA.

6.On a coloured paper draw an angle of 90 degrees and cut it. Name the angle as DEF.

7.Place angle DEF on PQ such that EF and AQ coincides.

OBSERVATION:We observe that angle OAQ is completely covered by angle DEF.This shows angle OAQ is a right angle .Thus,OA is prependicular to PQ where OA is radius and PQ is tangent.

Result :Radius of a circle is perpendicular to the tangent through the point of contact.

angle in a segment

AIM By Paper cutting and pasting verify the following

A)angle in a minor segment is obtuse .

B)angle in major segment is acute.


MATERIAL REQUIRED-

Coloured paper, fevistick, scissors, carbon paper, geometry box.


PROCEDURE-(A)

1. Draw a circle of any radius on a coloured paper and paste it on a plane sheet.



2.Fold the circle in such a way that a chord AB is obtained, draw AB.



3.Take a point P in the minor segment.


4.Form a crease joining AP , draw AP.



5.Form a crease joining BP, draw BP.

6. Draw a right angle DEF on a paper.



6.Make a replica of angle APB.



7.Place the replica of angle APB on the right angle DEF such that BP falls on EF.




OBSERVATION(A)-

1.Angle APB does not cover angle DEF completely.


2.Angle APB is bigger than angle DEF.


3. Measure of angle APB is more than 90 degrees.


4.Angle APB is obtuse.


RESULT(A) - Angle in the minor segment is obtuse.


PROCEDURE-(B)


1.Draw a circle of any radius on a plane sheet.


2.Fold the circle in such a way that a chord AB is obtained.Draw AB.


3.Take a point Q in the major segment.


4.Form a crease joining AQ. Draw AQ.


5.Form a crease joining BQ.Draw BQ.


6.Make a replica of angle AQB.


7.Place the replica of angle AQB on a right angle DEF such that BQ falls on EF. OBSERVATION :(B )
Angle AQB is less than Angle DEF.

Result :Angle in major segement is acute.

angle in a semicircle

Aim: By paper cutting, pasting and folding verify that angle in a semi-circle is right angle.

Material Required: coloured paper, pair of scissors, glue, carbon paper,geometry box.

Procedure:

1:Draw a circle of any radius with centre 'O' on a coloured paper and cut it.

2:Paste the circle on a plane sheet.

3:Form a crease passing through the centre 'O' of thce circle such that diameter AB is obtained.

4:Take a point 'P' on the semicircle.

5:Form a crese joining AP & draw AP.

6: Form a crease joining BP and draw BP.

7:Make two replicas of angle APB using a carbon paper.

8: Draw a straight line and place replicas of angle APB adjacent to each other such that PB is common side.

Observations:

1:AB is the diameter of the circle.

2: Angle APB is angle in a swmicircle.

3: Replicas of angle APB,when placed adjacent to each other on a straight line, completely cover the straight angle.

4. Twice angle APB = 180 degrees

Result: Angle in the semi circle is a right angle.