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Part A) Proof
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Part A) Proof

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MA301 Assignment: Proof for property of circle

MA301 Assignment: Proof for property of circle

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  • 1. Method One
  • 2. Let Angle AOB be x°.
    We draw a line from O to P.
    Since lines OP, OB and OA are radii of the same circle, they are equal in length.
    From that, we can conclude that triangles OBP and OAP are isosceles triangles.
    P
    O
    B
    A
  • 3. Let Angle BOP be y°.
    Angle OBP = Angle OPB = (180°-y°)/2
    ------------- Sum of interior angles of a triangle is 180°
    ------------- Property of isosceles triangle
    P
    O
    B
    A
  • 4. Angle AOP = x°+y°
    Angle OAP = Angle OPA = (180°-x°-y°)/2
    ------------- Sum of interior angles of a triangle is 180°
    ------------- Property of isosceles triangle
    P
    O
    B
    A
  • 5. Angle APB = Angle OPB – Angle OPA
    = [(180°-y°)/2] – [(180°-x°-y°)/2]
    = (180°-y°-180°+x°+y°)/2
    = x°/2
    P
    O
    B
    A
    Therefore, this proves that the angle at the centre (Angle AOB = x°) is twice the angle at circumference (Angle APB = x°/2)
  • 6. Method TWO
  • 7. We draw a line form P to O and extend it out to C, forming the diameter of the circle.
    Therefore OC=OC=OB=OP since they are all radii of the circle.
    Thus triangles OCA, OAP and OBP is isosceles.
    P
    O
    C
    B
    A
  • 8. Let angle OBP be of value x°.
    Since triangle OBP is an isosceles triangle, angle OBP=angle OPB=x°
    Therefore angle COB= 2x°.
    ------ Properties of exterior angles
    P
    O
    C
    B
    A
  • 9. Let angle OAP be of value y°.
    Since triangle OAP is an isosceles triangle, angle OAP=angle OPA=y°
    Therefore angle COA= 2y°.
    ------ Properties of exterior angles
    P
    O
    C
    C
    B
    A
  • 10. Angle AOB= angle COB-angle COA
    = 2x°-2y°
    = 2(x-y)°
    Angle APB= angle OPB-angle OPA
    = x°- y°
    = (x-y)°
    P
    O
    C
    B
    A
    Therefore, this proves that the angle at the centre [Angle AOB = 2(x-y)°] is twice the angle at circumference [Angle APB = (x-y)°].

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