circle properties.ppt

8/9/2019 Circle Properties.ppt

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Properties

Circles

Circle Theorems Angles Subtended on the Same Arc

Angle in a Semi-Circle with Proof

Tangents

Angle at the Centre with Proof Alternate Segment Theorem with Proof

Cyclic Quadrilaterals


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Circles

A circle is a set of points

which are all a certaindistance from a fixed point

known as the centre (!"

A line #oining the centre of

a circle to any of the pointson the circle is known as a

radius (A $ %!"


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Circles

The circumference of a

circle is the length of thecircle" The circumferenceof a circle & ' ) theradius"

The red line in thisdiagram is called a chord"*t di+ides the circle into ama#or segment and aminor segment"


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Angles Subtended on the Same Arc

Angles formed from two

points on thecircumference are e,ual to

other angles in the same

arc formed from those two

points"


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Angle in a Semi-Circle

Angles formed by drawinglines from the ends of thediameter of a circle to itscircumference form a rightangle" So C is a rightangle"


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Proof

.e can split the triangle in two

by drawing a line from the

centre of the circle to the point

on the circumference our

triangle touches"

.e know that each of the lines

which is a radius of the circle(the green lines! are the same

length" Therefore each of the

two triangles is isosceles and

has a pair of e,ual angles"


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Proof

%ut all of these angles

together must add up to/012 since they are the

angles of the original big

triangle"

Therefore x 3 y 3 x 3 y &/012 in other words '(x 3

y! & /012" And so x 3 y &

412"


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Tangents

A tangent to a circle is a

straight line which touchesthe circle at only one point

A tangent to a circle forms

a right angle with the

circle5s radius at the pointof contact of the tangent"


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Tangents

*f two tangents are drawn

on a circle and they crossthe lengths of the two

tangents (from the point

where they touch the circle

to the point where theycross! will be the same"


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Angle at the Centre

The angle formed at thecentre of the circle by linesoriginating from two pointson the circle5scircumference is doublethe angle formed on thecircumference of the circleby lines originating fromthe same points" i"e" a &'b"


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Proof

A & 6 since both of

these are e,ual to theradius of the circle"

The triangle A6 is

therefore isosceles and so

6A & a∠ " Similarly6% & b∠ "


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Proof

Since the angles in a triangleadd up to /012 we know that

6A & /01∠ 2 - 'a Similarly %6 & /01∠ 2 - 'b Since the angles around a

point add up to 7812 we ha+e

that A% & 781∠ 2 - 6A -∠%6∠ & 7812 - (/012 - 'a! - (/012 -

'b! & 'a 3 'b & '(a 3 b! & ' A6%∠


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Alternate Segment Theorem

The alternate segment theorem shows that thered angles are e,ual to each other and thegreen angles are e,ual to each other"


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Proof

A tangent makes an angle

of 412 with the radius of acircle so we know that

AC 3 x & 41∠ 2"

The angle in a semi-circle

is 412 so %CA & 41∠ 2"


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Proof

The angles in a triangle

add up to /012 so %CA∠

3 AC 3 y & /01∠ 2"

Therefore 412 3 AC 3 y &∠/012 and so AC 3 y & 41∠ 2"

%ut AC 3 x & 412 so AC∠3 x & AC 3 y"∠

9ence x & y"


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Cyclic Quadrilaterals

A cyclic ,uadrilateral is a four-sided figure in a

circle with each +ertex (corner! of the,uadrilateral touching the circumference of the

circle" The opposite angles of such a

,uadrilateral add up to /012"

circle properties.ppt

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