1. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty1 Circles Revision of Angle Properties Angles in a semi-circle www.mathsrevision.com Pythagoras Theorem (S O H)(C A H)(T O A) Tangent line on a circle National 4 Isosceles Triangles in Circles

2. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty2 Starter Questions Starter Questions www.mathsrevision.com 34 o A B C 6 8 coco National 4

3. 8-Jun-15Created by Mr. Lafferty Learning Intention Success Criteria 1.To know the basic properties for angles. 1.To review angle properties. 2.Solve problems using properties. www.mathsrevision.com Revision Angle Properties National 4

4. 8-Jun-15Created by Mr. Lafferty 120 o 95 o www.mathsrevision.com Angles round a point Add up to 360 o 115 o Two angles making a straight line add to 180 o angles opposite each other at a cross are equal. 34 o 3 angles in a triangle ALWAYS add up to 180 o. 50 o 40 o 65 o 90 o 146 o 146 o 145 o Revision Angle Properties National 4

5. 8-Jun-15Created by Mr. Lafferty www.mathsrevision.com Two angles in a isosceles Are equal ALL angles in an equilateral triangle are 60 o d = 115 o aoao coco bobo gogo fofo hoho eoeo is corresponding to d and must be 115 o is opposite to d and must be 115 o is must be 65 o (straight line) is alternate to c and must also be 65 o h b c e Revision Angle Properties National 4

6. 8-Jun-15Created by Mr. Lafferty Now try Ex 1 Ch12 (page 135) Copy out shapes www.mathsrevision.com Revision Angle Properties National 4

7. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty7 Starter Questions Starter Questions www.mathsrevision.com 136 o A B C 3 4 National 4

8. 8-Jun-15Created by Mr. Lafferty Learning Intention Success Criteria 1.To identify isosceles triangles with circles. 1.To investigate isosceles triangles with circles. 2.Solve problems using angle properties. www.mathsrevision.com Revision Angle Properties National 4

9. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty9 When two radii are drawn to the ends of a chord, An isosceles triangle is formed. C AB www.mathsrevision.com Isosceles triangles in Circles x o x o DEMO

10. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty10 Special Properties of Isosceles Triangles Two equal lengths Two equal angles Angles in any triangle sum to 180 o www.mathsrevision.com Isosceles triangles in Circles

11. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty11 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 www.mathsrevision.com Isosceles triangles in Circles

12. 8-Jun-15Created by Mr. Lafferty Now try Ex 1 Ch12 (page 137) Copy out shapes www.mathsrevision.com Revision Angle Properties National 4

13. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty13 Starter Questions Starter Questions www.mathsrevision.com A B C 5 12 bobo 27 o National 4

14. 8-Jun-15Created by Mr. Lafferty Learning Intention Success Criteria 1.To know the special angle in a semi-circle. 1.To investigate the special angle in a semi-circle. 2.Solve problems using the special property. www.mathsrevision.com Angles in a Semi Circle National 4

15. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty15 Tool-kit required 1.Protractor 2.Pencil 3.Ruler www.mathsrevision.com Angles in a Semi Circle

16. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty16 1.Using your pencil trace round the protractor so that you have semi-circle. 2.Mark the centre of the semi-circle. You should have something like this. www.mathsrevision.com Angles in a Semi Circle

17. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty17 Mark three points 1. Outside the circle x x x x x x x x x 2. On the circumference 3. Inside the circle www.mathsrevision.com Angles in a Semi Circle

18. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty18 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 Log your results in a table. InsideCircumferenceOutside www.mathsrevision.com Angles in a Semi Circle

19. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty19 Inside Circumference Outside x x x < 90 o > 90 o = 90 o www.mathsrevision.com DEMO Angles in a Semi Circle

20. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty20 KeyPoint for Angles in a Semi-circle Angles in a Semi-Circle A triangle APB inscribed within a semicircle with hypotenuse equal to the diameter will ALWAYS be right angled at P on the circumference. Remember - Angles in any triangle sum to 180 o www.mathsrevision.com P AB National 4

21. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty21 Hints www.mathsrevision.com Example 1 : Sketch diagram and find all the missing angles. 70 o Look for right angle triangles 43 o 20 o 47 o Remember ! Angles in any triangle sum to 180 o Angles in a Semi-Circle National 4

22. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty22 www.mathsrevision.com Example 2 : Sketch the diagram. Angles in a Semi-Circle C AB D 25 o 60 o (a)Right down two right angle triangles (a)Calculate all missing angles. E National 4

23. 8-Jun-15Created by Mr. Lafferty Now try Ex 2 Ch12 (page 138) Sketch shapes www.mathsrevision.com Angles in a Semi-Circle National 4

24. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty24 Starter Questions Starter Questions www.mathsrevision.com National 4

25. 8-Jun-15Created by Mr. Lafferty Learning Intention Success Criteria 1.To know when to use Pythagoras Theorem in a circle. 1. To explain how we can use Pythagoras Theorem to calculate length within a circle. 2.Solve problems using Pythagoras Theorem. www.mathsrevision.com Angles in a Semi Circle Pythagoras Theorem National 4

26. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty26 www.mathsrevision.com Angles in a Semi-Circle Pythagoras Theorem We have been interested in right angled triangles within a semi-circle. Since they are right angled we can use Pythagoras Theorem to calculate lengths. P AB 4cm 3cm d cm Example 1 : Calculate the value of d 5cm National 4

27. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty27 www.mathsrevision.com Angles in a Semi-Circle Pythagoras Theorem Y XZ cm 8cm 10 cm Example 2 : Calculate the length of XY 6cm National 4

28. 8-Jun-15Created by Mr. Lafferty Now try Ex 3 Ch12 (page 140) Q1 to Q9 Sketch shapes www.mathsrevision.com Angles in a Semi-Circle Pythagoras Theorem National 4

29. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty29 Starter Questions Starter Questions www.mathsrevision.com S O HC A HT O A National 4

30. 8-Jun-15Created by Mr. Lafferty Learning Intention Success Criteria 1.To know when to use Trigonometry (S O HC A HT O A) Trigonometry (S O HC A HT O A) to calculate length and angles to calculate length and angles within a circle. within a circle. 1. To explain how we can use Trigonometry (S O HC A HT O A) to calculate length and angles within a circle. 2.Solve problems using Trigonometry (S O HC A HT O A). www.mathsrevision.com Angles in a Semi Circle S O HC A HT O A National 4

31. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015 Created by Mr Lafferty 31 www.mathsrevision.com National 4 Angles in a Semi-Circle We have been interested in right angled triangles within a semi-circle. Since they are right angled we can use SOHCAHTOA to calculate lengths and angles. Example 1 : Calculate the value of angle x o P AB 4cm 3cm d cm xoxo =36.9 o S O HC A HT O A

32. Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty32 www.mathsrevision.com Angles in a Semi-Circle B AC 67.4 o 13 cm Example 2 : Calculate the length of AB 5cm S O HC A HT O A National 4

33. 8-Jun-15Created by Mr. Lafferty Now try Ex 3 Ch12 (page 140) Q10 onwards Sketch shapes www.mathsrevision.com Angles in a Semi-Circle S O HC A HT O A

34. National 4 If a = 7 b = 4 and c = 10 Write down as many equations as you can Starter Questions Starter Questions Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty34 www.mathsrevision.com e.g.a + b + c = 21

35. National 4 8-Jun-15Created by Mr. Lafferty Learning Intention Success Criteria 1.To understand what a tangent line is. 1.To explain how what a tangent line is and its special property with the radius at the point of contact. 2.Solve problems using the tangent property. www.mathsrevision.com Angles in a Semi Circle Tangent Line

36. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty36 A tangent line is a line that touches a circle at only one point. Which of the lines are tangent to the circle? www.mathsrevision.com Angles in a Semi-Circle Tangent Line

37. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty37 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. www.mathsrevision.com Angles in a Semi-Circle Tangent Line DEMO

38. National 4 Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty38 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 www.mathsrevision.com Angles in a Semi-Circle Tangent Line

39. National 4 8-Jun-15Created by Mr. Lafferty Now try Work Sheet Sketch shapes www.mathsrevision.com Angles in a Semi-Circle Tangent Line