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The Circle

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  1. The Circle Isosceles Triangles in Circles Right angle in a Semi-Circle Tangent Line to a Circle Diameter Symmetry in a Circle Circumference of a Circle www.mathsrevision.com Length of an ARC of a Circle Area of a Circle Area of a SECTOR of a Circle Summary of Circle Chapter Created by Mr Lafferty

  2. Starter Questions Q1. True or false www.mathsrevision.com Q2. How many degrees in one eighth of a circle. Q3. Q4. After a discount of 20% an iPod is £160. How much was it originally. Created by Mr Lafferty

  3. Isosceles Triangles in Circles Aim of Today’s Lesson We are learning to identify isosceles triangles within a circle. www.mathsrevision.com Created by Mr Lafferty

  4. Isosceles triangles in Circles When two radii are drawn to the ends of a chord, an isosceles triangle is formed. DEMO A B xo xo www.mathsrevision.com C Created by Mr Lafferty

  5. Isosceles triangles in Circles Special Properties of Isosceles Triangles Two equal lengths www.mathsrevision.com Two equal angles Angles in any triangle sum to 180o Created by Mr Lafferty

  6. Solution Angle at C is equal to: Isosceles triangles in Circles Q. Find the angle xo. B www.mathsrevision.com C xo A Since the triangle is isosceles we have 280o Created by Mr Lafferty

  7. Isosceles triangles in Circles Maths in Action Ex 2.1 page 181 www.mathsrevision.com Created by Mr Lafferty

  8. Starter Questions Q1. Explain how we solve Q2. How many degrees in one tenth of a circle. www.mathsrevision.com Q3. Q4. After a discount of 40% a Digital Radio is £120. Explain why the originally price was £200. Created by Mr Lafferty

  9. Semi-circle angle Aim of Today’s Lesson We are learning 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. www.mathsrevision.com Created by Mr Lafferty

  10. Semi-circle angle Tool-kit required 1. Protractor www.mathsrevision.com 2. Pencil 3. Ruler Created by Mr Lafferty

  11. You should have something like this. Semi-circle angle 1. Using your pencil trace round the protractor so that you have semi-circle. 2. Mark the centre of the semi-circle. www.mathsrevision.com Created by Mr Lafferty

  12. Semi-circle angle x x x x • Mark three points • Outside the circle x x x 2. On the circumference x x 3. Inside the circle www.mathsrevision.com Created by Mr Lafferty

  13. Log your results in a table. Outside Circumference Inside Semi-circle angle x 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 www.mathsrevision.com DEMO Created by Mr Lafferty

  14. Outside Circumference Inside Semi-circle angle DEMO x x x = 90o > 90o < 90o www.mathsrevision.com Begin Maths in Action Book page 182 Created by Mr Lafferty

  15. Starter Questions If a = 7 b = 4 and c = 10 Write down as many equations as you can www.mathsrevision.com e.g. a + b = 11 Created by Mr Lafferty

  16. Tangent line Aim of Today’s Lesson We are learning to understand what a tangent line is and its special property with the radius at the point of contact. www.mathsrevision.com Created by Mr Lafferty

  17. Which of the lines are tangent to the circle? Tangent line A tangent line is a line that touches a circle at only one point. www.mathsrevision.com Created by Mr Lafferty

  18. Tangent line The radius of the circle that touches the tangent line is called the point of contact radius. DEMO Special Property The point of contact radius is always perpendicular (right-angled) to the tangent line. www.mathsrevision.com Created by Mr Lafferty

  19. Solution Right-angled at A since AC is the radius at the point of contact with the Tangent. Tangent line Q. Find the length of the tangent line between A and B. B 10 www.mathsrevision.com By Pythagoras Theorem we have C A 8 Created by Mr Lafferty

  20. Tangent line Maths in Action Ex 4.1 page 185 www.mathsrevision.com Created by Mr Lafferty

  21. Starter Questions Q1. Using FOIL multiply out (x2 + 4x - 3)(x + 1) Using a multiplication table expand out (x2 + 4x - 3)(x + 1) www.mathsrevision.com Q2. Q3. I want to make 15% profit on a computer I bought for £980. How much must I sell it for. Created by Mr Lafferty

  22. Diameter symmetry Aim of Today’s Lesson We are learning to understand some special properties when a diameter bisects a chord. www.mathsrevision.com Created by Mr Lafferty

  23. C A B D Diameter symmetry • A line drawn through the centre of a circle • through the midpoint a chord will ALWAYS cut • the chord at right-angles O • A line drawn through the centre of a circle • at right-angles to a chord will • ALWAYS bisect that chord. • A line bisecting a chord at right angles • will ALWAYS pass through the centre of a circle. DEMO Created by Mr Lafferty

  24. Diameter symmetry Q. Find the length of the chord A and B. Solution Radius of the circle is 4 + 6 = 10. B Since yellow line bisect AB and passes through centre O, triangle is right-angle. 10 O www.mathsrevision.com By Pythagoras Theorem we have 4 6 Since AB is bisected The length of AB is A Created by Mr Lafferty

  25. Diameter symmetry Maths in Action Ex 5.1 & Ex 5.2 page 187 www.mathsrevision.com Created by Mr Lafferty

  26. 8m 12m Starter Questions Q1. Q2. Q3. Explain why the area of the triangle is 48m2 www.mathsrevision.com Q4. Created by Mr Lafferty

  27. Circumference of a circle Aim of Today’s Lesson We are learning to use the formula for calculating the circumference of a circle www.mathsrevision.com Created by Mr Lafferty

  28. radius Diameter Circumference Circumference of a circle Main parts of the circle O www.mathsrevision.com Created by Mr Lafferty

  29. Solution Q.Find the circumference of the circle ? Circumference of a circle 4cm www.mathsrevision.com Created by Mr Lafferty

  30. Solution • The circumference of the circle is 60cm ? • Find the length of the diameter and radius. Circumference of a circle www.mathsrevision.com Created by Mr Lafferty

  31. Circumference of a circle Now it’s your turn ! Maths in Action Ex 7.1 page 191 www.mathsrevision.com Created by Mr Lafferty

  32. Starter Questions Q1. True or false Q2. Using the balancing method rearrange into D = www.mathsrevision.com Q3. Q4. Calculate Created by Mr Lafferty

  33. length of the arc of a circle Aim of Today’s Lesson We are learning to use the formula for calculating the length of an arc. www.mathsrevision.com Created by Mr Lafferty

  34. minor arc major arc Q. What is an arc ? Arc length of a circle Answer An arc is a fraction of the circumference. A www.mathsrevision.com B Created by Mr Lafferty

  35. Solution Q. Find the circumference of the circle ? Arc length of a circle 10cm www.mathsrevision.com Created by Mr Lafferty

  36. Q. Find the length of the minor arc XY below ? Arc length of a circle x Arc length Arc angle = connection πD 360o y 6 cm 45o www.mathsrevision.com 360o DEMO Created by Mr Lafferty

  37. Q. Find the length of the minor arc AB below ? Arc length of a circle Arc length Arc angle = connection A πD 360o 9 cm www.mathsrevision.com 60o B Created by Mr Lafferty

  38. Q. Find the length of the major arc PQ below ? Arc length of a circle Arc length Arc angle = connection P πD 360o 10 m www.mathsrevision.com 260o 100o Q Created by Mr Lafferty

  39. length of the arc of a circle Now it’s your turn ! Maths in Action Ex 8.1 page 193 www.mathsrevision.com Created by Mr Lafferty

  40. Starter Questions Q1. True or false Q2. Expand out (x + 3)(x2 + 40 – 9) www.mathsrevision.com Q3. Q4. I want to make 30% profit on a DVD player I bought for £80. How much must I sell it for. Created by Mr Lafferty

  41. The Area of a circle Aim of Today’s Lesson We are learning to use the formula for calculating the area of a circle www.mathsrevision.com Created by Mr Lafferty

  42. If we break the circle into equal sectors And lay them out side by side We get very close to a rectangle. 1 8 2 2 7 1 3 6 5 8 6 4 4 3 7 5 The Area of a circle www.mathsrevision.com Created by Mr Lafferty

  43. If we cut the sectors Thinner and thinner then we get closer and closer to a rectangle. Hence we can represent the area of a circle by a rectangle. 8 6 4 2 3 7 1 5 thinner and thinner sectors The Area of a circle www.mathsrevision.com Created by Mr Lafferty

  44. The Area of a circle www.mathsrevision.com But the area inside this rectangle is also the area of the circle Created by Mr Lafferty

  45. Solution Q.Find the area of the circle ? The Area of a circle 4cm www.mathsrevision.com Created by Mr Lafferty

  46. The Area of a circle Now it’s your turn ! Begin Maths in Action Book Ex9.1 page 194 Q1-2 www.mathsrevision.com Created by Mr Lafferty

  47. Solution • The diameter of the circle is 60cm. • Find area of the circle? The Area of a circle www.mathsrevision.com Created by Mr Lafferty

  48. The Area of a circle Now it’s your turn ! Maths in Action Ex9.1 page 194 www.mathsrevision.com Created by Mr Lafferty

  49. Solution • The area of a circle is 12.64 cm2. • Find its radius? The Area of a circle www.mathsrevision.com Created by Mr Lafferty

  50. The Area of a circle Now it’s your turn ! Begin Maths in Action Book Ex9.1 page 194 Q5 onwards www.mathsrevision.com Created by Mr Lafferty