Circle Properties Part I

A circle is a set of all points in a plane that are the same distance from a fixed point in a plane The set of points form the. Circumference

The line joining the centre of a circle and a point on the circumference is called the………………. Radius

A is a straight line segment joining two points on the circle chord

A chord that passes through the centre is a ………………………. diameter

A……………………… is a straight line that cuts the circle in two points secant

An arc is part of the circumference of a circle Major arc Minor arc

A ……………………is part of the circle bounded by two radii and an arc sector Minor sector major sector

A ……………………is part of the circle bounded by a chord and an arc segment Minor segment major segment

The arc AB subtends an angle of  at the centre of the circle. A B O  Subtends means “to extend under” or “ to be opposite to”

Instructions: Draw a circle Draw two chords of equal length Measure angles  AOB and  DOC A B C D O What do you notice?

  Equal chords subtend equal angles at the centre

  Conversely Equal angles at the centre of a circle stand on equal arcs

Instructions: select an arc AB subtend the arc AB to the centre O subtend the arc AB to a point C on the circumference Measure angles  AOB and  ACB B O A C What do you notice?

Instructions: select an arc AB subtend the arc AB to the centre O subtend the arc AB to a point C on the circumference Measure angles  AOB and  ACB B O A C What do you notice?

 22 The angle that an arc of a circle subtends at the centre is twice the angle it subtends at the circumference

Instructions: select an arc AB select two points C, D on the circumference subtend the arc AB to a point C on the circumference subtend the arc AB to a point D on the circumference Measure angles  ACB and  ADB B O A C D

Instructions: select an arc AB select two points C, D on the circumference subtend the arc AB to a point C on the circumference subtend the arc AB to a point D on the circumference Measure angles  ACB and  ADB B O A C D What do you notice?

  Angles subtended at the circumference by the same arc are equal  

Instructions: Draw a circle and its diameter subtend the diameter to a point on the circumference Measure  ACB C B What do you notice? A

An angle in a semicircle is a right angle

 γ Instructions: Draw a cyclic quadrilateral (the vertices of the quadrilateral lie on the circumference Measure all four angles β  What do you notice?

 180-  The opposite angles of a cyclic quadrilateral are supplementary  180- 

 180-  If the opposite angles of a quadrilateral are supplementary the quadrilateral is cyclic

β  Instructions: Draw a cyclic quadrilateral Produce a side of the quadrilateral Measure angles  and β

  If a side of a cyclic quadrilateral is produced, the exterior angle is equal to the interior opposite angle

Circle Properties Part IItangent properties

A tangent to a circle is a straight line that touches the circle in one point only

Tangent to a circle is perpendicular to the radius drawn from the point of contact.

Tangents to a circle from an exterior point are equal

When two circles touch, the line through their centres passes through their point of contact Point of contact External Contact

When two circles touch, the line through their centres passes through their point of contact Point of contact Internal Contact

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment  

The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point A B BA 2 =BC.BD C D B=external point

The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point A B BA 2 =BC.BD C D Note: B is the crucial point in the formula

Circle Properties Chord properties

A B C D X AX.XB=CX.XD Triangle AXD is similar to triangle CXB hence

A B C D X AX.XB=CX.XD Note: X is the crucial point in the formula

Chord AB and CD intersect at X Prove AX.XB=CX.XD A B C D X In  AXD and  CXB  AXD =  CXB (Vertically Opposite Angles)  DAX =  BCX (Angles standing on same arc)  ADX =  CBX (Angles standing on same arc)   AXD     CXB Hence (Equiangular ) AAA test for similar triangles

A B C A perpendicular line from the centre off a circle to a chord bisects the chord

A B C Conversley: A line from the centre of a circle that bisects a chord is perpendicular to the chord

A B C Equal chords are equidistant from the centre of the circle

A B C Conversley: Chords that are equidistant from the centre are equal

Quick Quiz

a 40  a= 40 

b b=b= 80  C

d 60  d=d= 120  C

f 55  f=f= C

m=m= 62 C 62  mm

e e=e= 90  C

x=x= 12 C 102  12 cm x cm

k 70  k=k= 35  C

a 120  a= 50  10 

x 100  x=x= 50  C

y y=y= 55  C 35 

Quick Quiz

answer= A Which quadrilateral is concyclic? A B C

c 60  c = 60  C Tangent

g g=g= 90  C Tangent

h=h= 4 C 4cm h cm Tangent

m 40  m = 50  C Tangent y = 50  y

a=a= 65 C 50  Q a P R PQ, RQ are tangents

n=n= 5 C n nx8=4x10 8n =40 n =5

q=q= 25 C 10 4 q 4q=10 2 4q=100 q=25

x=x= 12 C 8 4 x 4(4+x)=8 2 4(4+x)=64 4+x=16 x=12 BA 2 =BC.BD

k=k= 5 C 8m k 3m K 2 = K =5