The Circle Theorems
description
Transcript of The Circle Theorems
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Circle Theorems
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Circle TheoremRemember the basics:
Angles in a triangle sum to 1800
Angles on a line sum to 1800
Isosceles triangles (radius)
Angles about a point sum to
3600
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Parts of a Circle
Diame
ter
radius
chord
tangent
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400
800
THEOREM 1: ANGLE at the CENTRE of the CIRCLE is twice the angle at the circumference subtended by the same arc.
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THIS IS THE ONE MOST PEOPLE DONT SEE
1150
2300
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400800
LOOKS DIFFERENT BUT STILL
THE CENTRE
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SPECIAL CASE OF THE SAME RULE BUT MAKES A RULE BY ITSELF
900
1800
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THEOREM 2: Every angle at the circumference of a SEMICIRCLE, that is subtended by the diameter of the semi-circle is a right angle.
900
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RULE 3: Angles at the circumference in the same SEGMENT of a circle are equal
PALE BLUE AREA IS THE SEGMENT
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PALE BLUE AREA IS THE SEGMENT
RULE 3: Angles at the circumference in the same SEGMENT of a circle are equal
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THEOREM 3: Angles at the circumference in the same SEGMENT of a circle are equal
NOTE: Will lead you to SIMILAR triangles (one is an enlargement of the other)
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THEOREM 4: Opposite angles sum to 180 in a cyclic quadrilateral
CYCLIC QUADRILATERAL MUST touch the circumference at all four vertices
910
890
700
1100
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A tangent is a line that touches a circle at one point only. This point is called the point of contact.
A chord is a line that joins two points on the circumference.
chord
tangent
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Theorem 5 A tangent is perpendicular to a radius
radius
tangent
900
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Theorem 6 Tangents to a circle from the same point are equal in length
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Theorem 7 The line joining an external point to the centre of a circle bisects the angle between the tangents
700350
350
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Theorem 5&7 combined can help you find the missing angles..
700350
350
900
900
xy
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Theorem 8 A radius bisects a chord at 900
radius cho
rd
900
And the chord will be cut perfectly in half...
MIDPOINT OF THE CHORD
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Theorem 9 Alternate angle theoremNeed a tangent
And a triangle that joins the tangent and two other points on the circumference of the circle
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Theorem 9 Alternate angle theorem
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Theorem 9 Alternate angle theoremThe angle between a tangent and a chord
Is equal to the angle in the alternate segment
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Theorem 9 Alternate angle theoremThe angle between a tangent and a chord
Is equal to the angle in the alternate segment
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COMMON ERRORS
IS TO THINK THIS IS A
DIAMETER SO..
THIS MUST BE 900
TANGENT MEETS
RADIUS
IT IS ONLY A DIAMETER IF YOU ARE TOLD SO
READ QUSETIONS CAREFULLY..