Know all about a circle
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Transcript of Know all about a circle
THE COLLECTION OF ALL THE POINTS IN A PLANE , WHICH ARE AT A FIXED
DISTANCE FROM A FIXED POINT IN A
PLANE, IS CALLED A CIRCLE
Know all about a Circle
Parts of a circle
Click icon to add picture
O
A B
C
DLine OB and OA are the radii of the circle
AB and CD are chords of the circle
CF is also the chord of the cirle known as DIAMETER
Diameter is the longest ----------------- of the circle
F
Area in green part is known as major sector
Area in minor part is known as -----------------
And the arc comprised in these sectors are respectively known as
Major arc
Minor arc.
Angles made in circle : the angles lying anywhere ON the the circle made by chords is known as SUBTENDED angle ( line AC is the chord)
Angle ABC is subtended angle in circle with centre o
Angle DOE is the central angle as it is making angle at the centre.
Major segment , minor segment and Semicircles
A segment is any region in a circle separated by a chord
Portion in green region is known as the Major segment
Portion in purple color is known as minor segment
What is the segment separated by a diameter known as??
Quick recap of all the terms From the figure aside name the following :
1. Points in the interior of the circle
2. Diameter of the circle
3.Radius of the circle
4.Subtended angle in the circle
5.Central angle in the circle
6.Major sector
7.Minor sector
8.Semicricle
A
BO
CD
Equal chords of a circle subtend equal angles at the centre
Equal chords of a circle subtend equal angles at the centre
Click icon to add pictureGiven: Chord AB = chord DC
To Prove:
angle AOB= angle DOC
Proof:
In Triangle ABC and triangle DOC
AB=DC given
AO=OC radii of same circle
BO=OD radii of same circle
Triangle AOB= Triangle DOC
angle AOB= angle DOC (C.P.C.T)
Hence proved…….
O
A
BC
O
D
If the angles subtended by the chords of a circle at the centre are congruent , then the chords are congruent.
Click icon to add pictureGiven :
Angle AOB= angle COD
To prove:
chord AB= Chord CD
Proof:
In triangle AOB and triangle COD
Angle AOB= angle COD (given )
AO=OC radii of same circle
BO=OD radii of same circle
Triangle AOB= Triangle DOC
chord AB= Chord CD
A B`
C
D
O
The perpendicular from the centre of the circle bisects the chord.
Click icon to add pictureGiven :
OD perpendicular AB
To prove:AD=DB
Proof:
In triangle AOD and triangle DOB
OA=OB radius
OD=OD common side
Angle ODA=angle ODB (90 degrees.)
Triangle AOD=ODB
(R-H-S test)
AD=DB ( C.P.C.T)
O
A BD
The line drawn through the centre of a circle to bisect the chord is perpendicular to the chord
Click icon to add pictureGiven : AD=DB
To prove: OD perpendicular AB
Proof:
In triangle AOD and triangle DOB
OA=OB radius
OD=OD common side
AD=DB given
triangle AOD = triangle DOB S-S-S test
Angle ODB=OAD (C.P.C.T)
Angle ODB+angle OAD=180 linear pair
Angle ODB= ½ angleADB
Angle ODB=90
O
A BD
Circle through 1,2,3, points
On a sheet of paper try drawing circle through one point
Two points Three pointsWhat do you see?
Answers
Many circles can be drawn from one pointMany circles can be drawn from two points But one and only one circle can be drawn
from three points.
The length of the perpendicular from a point to a line is the (shortest) distance of the line from the centre
Click icon to add pictureTry naming them and proving it.
OD is perpendicular to the line
Others are all hypotenuse
In a right angle triangle hypotenuse is the longest side…
So ………………………………………….
O
D
Equal chords of a circle (or congruent circles) are equidistant from the centre
Click icon to add pictureGiven: AB=CD
To prove: OF=OE
Draw OF perpendicular to OE
OOOA
B
C
E
D
O
F
Pick statements in proper order to prove the theorem and match the reasons
Statements AF=FB
AF=1/2AB CE=ED
CE=1/2CD CE=AF
Chord AF=chord CEOA =OCOB=ODIn triangles AOF and OCETriangles congruent byAngle F= Angle EOF=OE
Reasons Radii of same circleC.P.C.TGivenRadii of same circle S-S-S test S-A-S testEach 90 degrees
Chords Equidistant from the centre of a circle are equal in length
(converse of the earlier theorem)
Try proving this…………………..
Have fun
Concentric circles : Circle with same centre are
known as concentric circ
ooo
Angles Subtended by an Arc of a chord.
Click icon to add pictureThe angels subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle
Angle . AMB is half of angle AOB
Angle AOB= angle of arc ACB
Angle AMB= ½ of arc AMB
o
A B
C
M
Angles in the same segment of a circle are equal
Click icon to add pictureAngles ADB
ACB
AEB
All lie in arc AMB
Hence all are equal to ½ arc AMB
So angle
ADB =ACB=AEB=1/2 arc AMB
AB
C
D
E
M
Cyclic Quadrilaterals
A Quadrilateral whose 4 corners are on sides of the circle is known as cyclic Quadrilateral
Properties of Cyclic Quadrilateral
1. the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees
If the sum of opposite angles of a quadrilateral is 180 degrees its cyclic quadrilateral.