Chapter 3 Circles Theorem notes F4 - Webs 3 Circles Theorem note… · Chapter 3 Circles Theorem...
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Circle Theorems Form 4 [email protected] 1 Chapter 3 Circles Theorem Section 3.1 Circle Properties v The CIRCUMFERENCE is the distance around the edge of a circle. v A CHORD is a straight line segment joining two points on a circle. v A DIAMETER is a chord that passes through the centre of a circle. v A RADIUS is the distance from the centre of a circle to a point on the circle. v A TANGENT is a line that touches the circle at only one point. v An ARC of a circle is any part of the circle's circumference. v A SECTOR is a region bounded by two radii and an arc lying between the radii. v A SEGMENT is a region bounded by a chord and an arc lying between the chord's endpoints
Transcript of Chapter 3 Circles Theorem notes F4 - Webs 3 Circles Theorem note… · Chapter 3 Circles Theorem...
CircleTheorems Form4
Chapter 3 Circles Theorem
Section 3.1 Circle Properties
v TheCIRCUMFERENCEisthedistance
aroundtheedgeofacircle.
v ACHORD is a straight line segment
joiningtwopointsonacircle.
v ADIAMETERisachordthatpassesthroughthecentreofa
circle.
v ARADIUS is the distance from the centre of a circle to a
pointonthecircle.
v ATANGENTisalinethattouchesthecircleatonlyonepoint.
v AnARCofacircleisanypartofthecircle'scircumference.
v ASECTORisaregionboundedbytworadiiandanarclyingbetweentheradii.
v A SEGMENT is a region bounded by a chord and an arc lying between the chord'sendpoints
CircleTheorems Form4
Section 3.2 Theorem 1 – The angle subtended by the arc at the centre of the circle is twice the angle subtended at the circumference
Subtended Angles
IfA,BandParethreepointsonthecircumferenceofacirclewithcentreO,thenwesaythat
AngleAPBissubtendedeitherbythearcABorbychordAB
Alternativelywecansaythat
both the arc AB and the chord AB subtend the angle APB at thecircumference
ThearcandchordalsosubtendtheangleAOBatthecentre.
Notethatachordcansubtendanobtuseangleatthecircumference.
Insuchacasetheangleisintheminorsegment.
The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference
Angle AOB = 2 × ACB
CircleTheorems Form4
The angle subtended at the centre of a circle by an arc is twice any angle subtended at thecircumferencebythesamearc
Example1
p=2x48
p=96(∠atcentre=2x∠atcirc)
Example2
p=2x115
p=230(∠atcentre=2x∠atcirc)
Example3
p=2x51
p=102(∠atcentre=2x∠atcirc)
CircleTheorems Form4
Example4Findthevalueofanglep.
Example5
Findthevalueofanglex.
SupportExercisePg481Exercise29CNo1
CircleTheorems Form4
Section 3.3 Theorem 2: Angles in the same segment are equal
Anglessubtendedbythesamearcareequal.Example1Findthemissinganglesgivingreasonsforyouranswer.
AngleACB=74(Lssubtendedbythesamechord)
Example2
Findthemissinganglesandgivereasonsforyouranswer.
a=41x2=82(∠atcentre=2x∠atcirc)
b=41(Lssubtendedbythesamechord)
AngleAPB=AngleAQB
CircleTheorems Form4
Example3
Findthemissinganglesandgivereasonsforyouranswer.
a=120/2=60(∠atcentre=2x∠atcirc)
b=60(Lssubbythesamechord)
c=32(Lssubbythesamechord)
SupportExercisePg481Exercise29CNo2,6,8
Handout
Section 3.4 Theorem 3: The angle subtended by the diameter is a right angle
Thiscanbederivedfromtheprevioustheorem.Sincetheangleatthecentre(180)istwicetheangleatthecircumference(90)wecansaythattheangleatthecircumferenceisarightangle.
Theanglesubtendedbythediameteris90°.
AngleACB=90°
CircleTheorems Form4
Example3
Findtheunknownangles
Example4
InthediagramXYisadiameterofthecircleand∠AZXisa.
Bensaysthatthevalueofais50° .
Givereasonstoexplainwhyheiswrong.
SupportExercisePg481Exercise29CNo3,7
CircleTheorems Form4
Section 3.5 Theorem 4: The sum of the opposite angles of a cyclic
quadrilateral is 180°
Aquadrilateralwhosevertices(corners)alllieonthecircumferenceofacircleiscalledacyclicquadrilateral.
Example1
Findthemissinganglegivingareasonforyouranswer.
AngleSPQ+AngleSRQ=180°
And
AnglePSR+AnglePQR=180°
CircleTheorems Form4
Example2
Findthemissinganglegivingareasonforyouranswer.
Example3
Findthemissinganglegivingareasonforyouranswer.
CircleTheorems Form4
Example4
Findtheanglesmarkedinletters.
SupportExercisePg481Exercise29CNo4,9
Handout
Section 3.6 Theorem 5: The angle between a radius and tangent form a right angle
A tangent to a circle is a line which just touches the circle.
Remember:
A tangent is always at right angles to the radius where it touches the circle.
CircleTheorems Form4
Example4
SupportExercisePg475Exercise29BNos1,2,3
Handout
Section 3.7 Theorem 6: Tangents from the same external points to a circle are equal in length
Iftwotangentsaredrawnonacircleandtheycross,thelengthsofthetwotangents(fromthepointwheretheytouchthecircletothepointwheretheycross)willbethesame.
The lengths of two tangentsdrawn from the sameexternalpointareequal.
CircleTheorems Form4
Example5
SupportExercisePg475Exercise29BNos5,6
Handout
Section 3.8 Theorem 7: Alternate Segment Theorem
The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.
This is the circle property that is the most difficult to spot. Look out for a triangle with one of its vertices (corners) resting on the point of contact of the tangent.
CircleTheorems Form4
The angle between a tangent and chord is equal to the angle made by that chord in the alternate segment.
In this diagram we can use the rule to see that the yellow angles are equal, and the blue angles are equal.
Example 1
Find the angles marked with letters.
∠x=60°[AlternateSegmentTheorem]
∠y=180°–60°–40°=80°[Anglesinatriangle]
Example2
