math proof: angle at circumference is twice angle at centre
DESCRIPTION
A mini project by JH301 Group C^^TRANSCRIPT
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How do we prove that the angle at the centre is twice the angle at the
circumference?
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OP
B
C
A
Y
With the given
diagram, we draw a
line OP and a line OC,
where AC is the
diameter of the
circle.
Since,
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OP
B
C
A
Y
We know that AO =
OC = OP = OB,
because the radii of
a circle are all equal.
We can then
identify all the
isosceles triangles.
Also,
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Now, we are ready to solve!
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O
B
C
A
Y
Let’s look at AOB.
AOB lies on the
diameter of the circle.
Therefore, we can say
that AOB = 180- ( COP
+ POB).
We call that equation 1.
1: AOB = 180 – ( COP + POB)
P
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O
B
C
A
Y
P
1: AOB = 180 – ( COP + POB)2: POB = 180 - 2 OPB
Now let’s look at angle POB.Angle POB is part of an isosceles triangle POB, where OPB = OBP .Therefore, we can say thatPOB = 180 – (OPB+ OBP)Or POB = 180 - 2 OPB
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1: AOB = 180 – ( COP + POB)2: POB = 180 - 2 OPB
Given the 2 equations, we’ll substitute equation 2 into equation 1.
ÐAOB = 180 – ( COP + 180 - 2 OPB) = 180 – 180 - COP + 2 OPB = 2 OPB - COP
And… TADA!!!We have our 3rd equation formed!
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O
B
C
A
Y
P
Getting back to our circle,
We know that
OPB = OPA + APB
Simple. No explanation
needed:D
And that will be equation
4!
3: AOB = 2 OPB - COP 4: OPB = OPA + APB
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3: AOB = 2 OPB - COP 4: OPB = OPA + APB
Of course then, the next step will be to substitute 4 into 3!
AOB = 2 (OPA + APB) - COP = 2 OPA + 2 APB - COP 2 OPA + 2 APB - COP + AOB = 0
And here, we have our equation 5!!!
5: 2 OPA + 2 APB - COP + AOB = 0
Alright, back to the circle.
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O
B
C
A
Y
P
We know that AOP is an isoceles triangle where OAP = OPA
Therefore, we can say that AOP + 2 OPA = 180However, we also know that AOP + COP = 180 (Angles on a straight line)
Thus, we can see that 2 OPA = COP (Which will be equation 6!)
*This is also a property of triangles, where the sum of 2 interior opposite s = the exterior .5: 2 OPA + 2 APB - COP + AOB = 0
6: 2 OPA = COP
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5: 2 OPA + 2 APB - COP + AOB = 06: 2 OPA = COP
Again, we substitute equation 6 into 5.
2 OPA + 2 APB - 2 OPA + AOB = 02 APB - AOB = 0
2 APB = AOB
And…..
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VIOLA!!!!
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We’ve proven that AOB is
twice APB, where AOB is an
at the centre, and APB is an
at the circumference.
Conclusion:
Angle at the centre is
twice angle at the
circumference.
O
B
C
A
Y
P
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A simple mathematical proof brought
to you by
JH301 Math Group C
Credits:
Mr Christopher Cheng
Miss Maureen Ng