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Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty1 Length of Arc in a Circles Area of.

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Presentation on theme: "Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty1 Length of Arc in a Circles Area of."— Presentation transcript:

1 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty1 Length of Arc in a Circles Area of a Sector in a Circle The Circle Symmetry in a Circle Diameter & Right Angle in a Circle Semi-Circle & Right Angle in a Circle Tangent & Right Angle in a Circle Summary of Circle Chapter Mix Problems including Angles

2 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty2 Starter Questions Q3. Q1. Simplify Q2. How many degrees in one eighth of a circle. Q4. After a discount of 20% an iPod is £160. How much was it originally.

3 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty3 Aim of Today’s Lesson To find and be able to use the formula for calculating the length of an arc. Arc length of a circle

4 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty4 Q. What is an arc ? A B Answer An arc is a fraction of the circumference. minor arc major arc Arc length of a circle

5 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty5 Q. Find the circumference of the circle ?Solution 10cm Arc length of a circle

6 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty6 Arc length of a circle Arc length πD Arc angle 360 o = Q. Find the length of the minor arc XY below ? 6 cm 45 o x y 360 o connection

7 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty7 Arc length of a circle Arc length πDπD Arc angle 360 o = Q. Find the length of the minor arc AB below ? 9 cm 60 o A B connection

8 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty8 Arc length of a circle Arc length πDπD Arc angle 360 o = Q. Find the length of the major arc PQ below ? 10 m 100 o A B connection 260 o

9 Int 2 24-Aug-14Created by Mr. Now try Ex 1 Ch6 MIA (page 60) Arc length of a circle

10 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty10 Starter Questions Q3. Q1. Solve Q2. How many degrees in one tenth of a circle. Q4. After a discount of 40% a Digital Radio is £120. How much was it originally.

11 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty11 Sector area of a circle Aim of Today’s Lesson To find and be able to use the formula for calculating the sector of an circle.

12 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty12 A B minor sector major sector Area of Sector in a circle

13 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty13 Q. Find the area of the circle ?Solution 10cm Area of Sector in a circle

14 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty14 Area Sector πr 2 Sector angle 360 o = Find the area of the minor sector XY below ? 6 cm 45 o x y 360 o connection Area of Sector in a circle

15 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty15 Area Sector πr2πr2 Sector angle 360 o = Q. Find the area of the minor sector AB below ? 9 cm 60 o A B connection Area of Sector in a circle

16 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty16 Area of Sector in a circle Sector Area πr2πr2 Sector angle 360 o = Q. Find the area of the major sector PQ below ? 10 m 100 o A B connection 260 o

17 Int 2 24-Aug-14Created by Mr. Now try Ex 2 Ch6 MIA (page 62) Area of Sector in a circle

18 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty18 Starter Questions Q3. Q1. Find the gradient and y - intercept Q2. Expand out (x 2 + 4x - 3)(x + 1) Q4. I want to make 15% profit on a computer I bought for £980. How much must I sell it for.

19 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty19 Arc length of a circle Arc length πDπD Arc angle 360 o = Q. The arc length is 7.07cm. Find the angle x o 9 cm xoxo A B connection

20 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty20 Area of Sector in a circle Sector Area πr2πr2 Sector angle 360 o = Q. Find the angle given area of sector AB is cm 2 ? 10 m A B connection x o

21 Int 2 24-Aug-14Created by Mr. Now try Ex 3 Ch6 MIA (page 64) Mixed Problems

22 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty22 Starter Questions Q3. Q1. Find the gradient and y - intercept Q2. Expand out (x + 3)(x – 9) Q4. I want to make 30% profit on a DVD player I bought for £80. How much must I sell it for.

23 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty23 To identify isosceles triangles within a circle. Aim of Today’s Lesson Isosceles triangles in Circles

24 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty24 When two radii are drawn to the ends of a chord, An isosceles triangle is formed. C A B Isosceles triangles in Circles Online Demo x o x o

25 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty25 Special Properties of Isosceles Triangles Two equal lengths Two equal angles Angles in any triangle sum to 180 o Isosceles triangles in Circles

26 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty26 Q.Find the angle x o. A B C Solution Angle at C is equal to: Since the triangle is isosceles we have xoxo 280 o Isosceles triangles in Circles

27 Int 2 24-Aug-14Created by Mr. Now try Ex 4 Ch6 MIA (page 66) Isosceles triangles in Circles

28 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty28 Starter Questions Q3. Q1. Find the gradient and y - intercept Q2. Expand out 2(x - 3) + 3x Q4. Car depreciates by 20% each year. How much is it worth after 3 years.

29 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty29 To understand some special properties when a diameter bisects a chord. Aim of Today’s Lesson Diameter symmetry

30 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty30 Diameter symmetry AB C D 1.A line drawn through the centre of a circle and through the midpoint a chord will ALWAYS cut the chord at right-angles 2.A line drawn through the centre of a circle at right-angles to a chord will ALWAYS bisect (cut equally in 2) that chord. 3.A line bisecting a chord at right angles will ALWAYS pass through the centre of a circle. o

31 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty31 Q.Find the length of the chord A and B. A B 6 O Solution By Pythagoras Theorem we have Diameter symmetry 4 Since yellow line bisect AB and passes through centre O, triangle is right-angle. Radius of the circle is = Since AB is bisected The length of AB is

32 Int 2 C Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty32 Diameter symmetry AB D o M Find the length of OM and CM. Radius of circle is 5cm, AB is 8cm and M is midpoint of AB. By Pythagoras Theorem we have AM is 8 ÷ 2 = 4 Radius of the circle is 5cm. CM = 3cm + radius CM = = 8cm

33 Int 2 24-Aug-14Created by Mr. Now try Ex 5 Ch6 MIA (page 68) Symmetry & Chords in Circles

34 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty34 Starter Questions Q3. Q1. Find the gradient and y - intercept Q2. Expand out (x + 3)(x + 4) Q4. Bacteria increase at a rate of 10% per hour. If there were 100 bacteria initial. How many are there after 9 hours.

35 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty35 Semi-Circle Angle To find the angle in a semi-circle made by a triangle with hypotenuse equal to the diameter and the two smaller lengths meeting at the circumference. Aim of Today’s Lesson

36 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty36 Semi-circle angle Tool-kit required 1.Protractor 2.Pencil 3.Ruler

37 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty37 1.Using your pencil trace round the protractor so that you have semi-circle. semi-circle. 2.Mark the centre of the semi-circle. You should have something like this. Semi-circle angle

38 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty38 Mark three points 1.Outside the circle x x x x x x x x x Semi-circle angle 2. On the circumference 3. Inside the circle

39 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty39 For each of the points Form a triangle by drawing a line from each end of the diameter to the point. Measure the angle at the various points. x x x Semi-circle angle Log your results in a table. InsideCircumferenceOutside

40 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty40 Inside Circumference Outside x Semi-circle angle x x < 90 o > 90 o = 90 o Begin Maths in Action Book page 182 Online Demo

41 Int 2 24-Aug-14Created by Mr. Now try Ex 7 Ch6 MIA (page 71) Angles in a Semi-Circle

42 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty42 Starter Questions Q3. Q1. Find the gradient and y - intercept Q2. Expand out (a - 3)(a 2 + 3a – 7) Q4. I want to make 60% profit on a TV I bought for £240. How much must I sell it for.

43 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty43 To understand what a tangent line is and its special property with the radius at the point of contact. Aim of Today’s Lesson Tangent line

44 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty44 Tangent line A tangent line is a line that touches a circle at only one point. Which of the lines are tangent to the circle?

45 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty45 Tangent line The radius of the circle that touches the tangent line is called the point of contact radius. Special Property The point of contact radius is always perpendicular (right-angled) to the tangent line. Online Demo

46 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty46 Tangent line Q.Find the length of the tangent line between A and B. A and B. A B 8 10 C Solution Right-angled at A since AC is the radius at the point of contact with the Tangent. By Pythagoras Theorem we have

47 Int 2 24-Aug-14Created by Mr. Now try Ex 8 Ch6 MIA (page 73) Tangent Lines

48 Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty48 Area is Summary of Circle Topic Circumference is Sector area Arc length is Diameter Radius line that bisects a chord 1.Splits the chord into 2 equal halves. 2.Makes right-angle with the chord. 3.Passes through centre of the circle Pythagoras Theorem SOHCAHTOA Semi-circle angle is always 90 o Tangent touches circle at one point and make angle 90 o with point of contact radius


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