The perpendicular bisector of a chord passes through the centre of a circle

2360

280

620

BA

O

M

OBM = 280 Prove this.

AOB = 3600 -2360 = 1240

AOB is isosceles Why?

OBM = ½ x (180-124) = 280

MOB = 620 Why?

OMB = 180 – (62+28) = 900

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Great Marlow School Mathematics Department

6cm

6cm O

C

M

A

4.25

cm

M is the mid point of AC.

Find the length of AC and OM.

Use pythagoras to find AC.

AC2 = 62 + 62

AC2 = 36 + 36 = 72

AC = 8.5 (1 dp)

CM = 4.25

OM is perpendicular to AC. Explain!

The perpendicular bisector of a chord passes through the centre of a circle

OM2 = OC2 – CM2

OM2 = 62 – 4.252

OM2 = 17.94 (2 dp)

OM = 4.24 (2.dp)Great Marlow School Mathematics Department

The tangent to a circle meets a radius at right angles.

We need to prove that PMA and PMB are congruent.

B

A

P

O

PM is common

PMB = PMA

AM = BM

PMB = PMA

(SAS)

PB = PA

- M

^^

Tangents from the same point are equal.

Great Marlow School Mathematics Department

A

O

B

C

28 0

x

zy

w

t

AB is a diameter of a circle.

ACB is an angle in a semicircle.

OC is a radius.

CBO = 280

OC = OB = OA Why?

x = 280 Why?

y = 1240 Why?

z = 560 Why ?

w = 620 t = 620 Why?

x + t = 280 + 620 = 900

ACB = 900

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The angle in a semicircle equals 900.

Great Marlow School Mathematics Department

Great Marlow School Mathematics Department

A B

O

C

D

AOB = 2 x ACB Prove this.

w + z = x + y + z (= 1800)

-z -z

w = x + y

x = y Reason?

w = y + y

w = 2 y

x

y

z

w s

t

u

v

s + t = t + u + v (=1800)

-t -t

s = u + v

u = v

s = v + v

s = 2v

w = 2y

s = 2v

w + s = 2y + 2v = 2(y + v)

AOB = 2 x ACB

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The angle at the centre is twice the angle at the circumference.

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A B

O

C

D

x

y

z

w st

u

v

Great Marlow School Mathematics Department

x y

z

O

z = 2x

z = 2y

2x = 2y

x = y

Prove x = y

A B

Angles at the circumference standing on the same arc are equal.

Great Marlow School Mathematics Department

a + b = 1800

c + d = 1800

a

b

d

c

Opposite angles in a cyclic quadrilateral add up to 1800.

Great Marlow School Mathematics Department

Find the angles marked. O is the centre of the circle.

1

O960

654

32

OO

O

O

Oa

d

c

b

230

170

570

d

880

nm

1100950

j

h

760

e

g

f

700

Great Marlow School Mathematics Department

Question 1

A, B, C and D are points on the circumference of a circle centre O.AC is a diameter of the circle.Angle BDO = x°.Angle BCA = 2x°.Express, in terms of x, the size ofi) angle BDA,ii) angle AOD,iii) angle ABD. (4 marks) [4] Total = 4

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-----------------------------------------Question 1Two tangents are drawn from a point T to a circle centre O.They meet the circle at points A and B.Angle AOB is equal to 128º.

In this question you MUST give reasons for your answers.Work out the size of the anglesi) APBii) BAOiii) ABT [5] Total = 5

Great Marlow School Mathematics Department