circle theorem powerpoint
TRANSCRIPT
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Circle Theorem
Remember to look for “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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Name parts of a circle
Diameter
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
MUST BE THE
CENTER
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This rule can be hard to spot…..
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THIS IS THE ONE MOST PEOPLE DON’T SEE......
1150
2300
MUST BE THE
CENTER
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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 IN ITS OWN RIGHT!!
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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THEOREM 3: Opposite angles sum to 180 in a cyclic quadrilateral
CYCLIC QUADRILATEARAL MUST touch the circumference at all four vertices
910
890
700
1100
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Now have a go at Worksheet 1
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RULE 4: 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 4: Angles at the circumference in the same SEGMENT of a circle are equal
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THEOREM 4: 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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Enter the world of tangents and chords…..• 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
radiuscho
rd
900
And the chord will be cut perfectly in half!!!
MIDPOINT OF THE CHORD
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Have a go at worksheet 2
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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 EXAM ERROR!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..