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# Circular measure - PowerPoint PPT Presentation

Circular measure. Definition of p Definition of radians. Unit 4:Mathematics. Aims Introduce radians and circular theorem. Objectives Identify parts of a circle and calculate triangles within a circle. Calculate circular and segment measures. Re-Call.

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## PowerPoint Slideshow about ' Circular measure' - cathleen-lang

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### Circular measure

Definition of p

Aims

• Introduce radians and circular theorem.

Objectives

• Identify parts of a circle and calculate triangles within a circle.

• Calculate circular and segment measures.

• sine x = (side opposite x)/hypotenuse cosine x = (side adjacent x)/hypotenuse

tangent x =(side opposite x)/(side adjacent x)

• sin A = a/c, cosine A = b/c, & tangent A = a/b.

• The reciprocal ratios are trigonometric ratios, too.  They are outlined below.

• cotangent x = 1/tan x = (adjacent side)/(opposite side)

• secant x = 1/cos x = (hypotenuse)/(adjacent side)

• cosecant x = 1/sin x = (hypotenuse)/(opposite side)

• Take any size of circle.

Cis the Circumference,

the distance around the outside.

c

d

d is the Diameter.

• Cis the Circumference, the distance around the outside.

• d is the Diameter

C

1

2

3

d

C ynymwneud3gwaithd

1

2

3

d

C = 3.1415927 x d

=3.1415927 (pi)

This is true for any size of circle

s

 =

r

+

s

r

Calculate the measure of the arc length below?s in the circle pictured below?

Definition below?

S

 =

r

r

S = 2pr

• Circumference below?

• The circumference of a circle is the perimeter of the circle

GylcheddCircumference

• The below?diameter of a circle is a line across the circle which passes through the centre.

• The radius of a circle is the distance from the centre of the circle to any point on the circumference. The radius is half the length of the diameter

• Segment below?

• A chord divides a circle into two segments: a minor segment and a major segment.

Chord

minor

Segment

bach

Tant chord

major

Segment

mawr

Tangent below?

• A tangent is a line which touches the circumference of a circle at one point only and is parallel to the circumference at that point.

• An arc is part of the circumference of a circle.

• A sector is formed between 2 radii and the circumference

arc

tangent

sector

Properties below?of a Circle

• The angle in a semi-circle is always a right angle

• If 2 chords are drawn from a point on the circumference of a circle to each end of a diameter the angle between the two chords is always a right angle.

• The below?angle at the centre of a circle

= twice the angle at the circumference

If lines are drawn from each end of a chord to a point on the circumference of a circle and to the centre, the angle at the centre is twice the angle at the circumference.

• Angles below?in the same segment are equal

If two chords are drawn from a point on the circumference of a circle to each end of a third chord the intersecting angle is the same no matter where the point is providing the points are in the same segment of the circle.

Opposite below?angles in a cyclic quadrilateral are supplementary

A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle.

Tangents below?

• A radius drawn from the point where a tangent touches a circle is perpendicular (at 90) to the tangent.

• Tangents below?drawn to a circle from the same point outside the circle are equal in length. PA = PB

• OP bisects angle APB

A

P

O

B

• B below?is the centre of the circle. Find angles BPC, BCP, ABP and PAB.

• BP = BC,  BPC = BCP

• (isosceles triangle)

• BPC + BCP = 180 – 66 = 114

• BPC = BCP = 57

• ABP = 180 – 66 = 114

• (angles on a straight line = 180)

• PAB = ½ × 66 = 33

• (angle at centre of a circle = twice that at the circumference)

• P

66

A

C

B

• Calculate below?angles p, q and r.

p + 85 = 180 (opposite angles in a cyclic quadrilateral)

• p = 180 – 85 = 95

• q + 101 = 180 (opposite angles in a cyclic quadrilateral)

• q = 180 – 101 = 79

• r + q = 180 (angles on a

• straight line)

• r = 180 – 79 = 101

r

q

p

85

101

• 3. below?Work out angles a, b, c and d.

• a + 41 + 101 = 180 (angles in a triangle)

• a = 180 – 41 – 101 = 38

• b = 101 (opposite angles)

• d = 41 (angles in the same segment)

• c = 38

• (angles in thesamesegment)

• a

41

101

b

c

d

• 4. below?Work out angles a, b and c.

• c = 90 (angles in a semicircle)

• a + 59 + 90 = 180 (angles in a triangle)

• a = 180 – 59 – 90 = 31

• a + b = 90 (radius is

• perpendicular to the tangent)

• b = 90 – 31 = 59

c

59

b

a

P below?

• 5. Calculate angles XPY and OXY .

• PX = PY  PYX =  PXY

• (isosceles triangle)

• PYX = 75

• PXO = 90 (radius is

• perpendicular to the tangent)

• OXY = 90 – 75 = 15

• XPY = 180 – 75 – 75 = 30

• (angles in a triangle)

75

X

Y

O

• 6. below?XTY is a tangent to the circle, centre O. P and Q are points on the circumference. OQ is parallel to PT. Angle QOT = 37. Find angles OPT and PTY.

• OTP = 37 (alternate angles)

• OP = OT  OPT = OTP = 37(isosceles triangle)

• OTY = 90 (radius is

• perpendicular to the tangent)

• PTY = OTY - OTP

• PTY = 90 – 37 = 53

O

37

Q

P

T

Y

X

7. below?PTR is a tangent to the circle, centre O. The chord AB is parallel to PR. X is a point on the circumference. Angle ORT = 18.

• Work out angle AXB.

• OTR = 90 (radius is perpendicular to the tangent)

• TOR = 180 – 90 – 18 = 72 (angles in a triangle)

• TAB = ½ × 72 = 36 (angle at circumference = ½ angle at centre).

• ATP = 36 (alternate angles)

• ATO = 90 – 36 = 54

X

O

B

A

18

R

T

P

• below?OTR = 90, TOR = 72, TAB = 36,

• ATP = 36, ATO = 54

• OT = OB  OTB = OBT

• (isosceles triangle)

• OTB + OBT = 180 - TOR = 180 – 72 = 108

• OTB = ½ × 108 = 54

• ATB = ATO + OTB

• ATB = 54 + 54 = 108

• AXB = 180 – 108 = 72

• (angles in a cyclic

• X

O

B

A

18

R

T

P