The Circle

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

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

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

5. 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

6. 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. www.mathsrevision.com Created by Mr Lafferty

7. 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

8. 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. Created by Mr Lafferty

9. 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

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

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

12. 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

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

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

15. Q. Find the arc length for the semi-circle ? length of the arc of a circle Solution www.mathsrevision.com 180o Created by Mr Lafferty

16. Solution Q. Given part of the circle below. Find the arc length ? length of the arc of a circle 10cm 90o www.mathsrevision.com Created by Mr Lafferty

17. length of the arc of a circle 3 step process to calculating the length of an arc. Step 1 Find the circumference of the circle. Step 2 Find fraction of the whole circle. www.mathsrevision.com Step 3 Find the arc length. Created by Mr Lafferty

18. Solution Q.Part of a circle is given below. Find the arc length. length of the arc of a circle Fraction of circle 6cm 45o www.mathsrevision.com Created by Mr Lafferty

19. length of the arc of a circle 3 step process to calculating the length of an arc. Step 1 Find the circumference of the circle. Step 2 Find fraction of the whole circle. www.mathsrevision.com Step 3 Find the arc length. Created by Mr Lafferty

20. Solution Q. Find the length of the arc AB below ? length of the arc of a circle A 9 cm www.mathsrevision.com 60o B Created by Mr Lafferty

21. 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

22. 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

23. minor sector major sector Sector area of a circle Q. What is an sector area ? Answer Sector area is a fraction of the area of the whole circle. A www.mathsrevision.com B Created by Mr Lafferty

24. Solution Q. Find the area of the circle ? Sector area of a circle 10cm www.mathsrevision.com Created by Mr Lafferty

25. Sector area of a circle Q. Find the sector area for the semi-circle ? Solution 10cm www.mathsrevision.com 180o Created by Mr Lafferty

26. Solution Sector area of a circle Q. Given part of the circle below. Find the sector area ? www.mathsrevision.com Created by Mr Lafferty

27. Sector area of a circle 3 step process to calculating sector area of a circle. Step 1 Find the area of the circle. Step 2 Find fraction of the whole circle. www.mathsrevision.com Step 3 Find the sector area. Created by Mr Lafferty

28. Solution Sector area of a circle Q.Part of a circle is given below. Find the sector area. Fraction of circle 6cm 45o www.mathsrevision.com Created by Mr Lafferty

29. Sector area of a circle 3 step process for calculating the sector area. Step 1 Find the area of the circle. Step 2 Find fraction of the whole circle. www.mathsrevision.com Step 3 Find the sector area. Created by Mr Lafferty

30. Solution Sector area of a circle Q. Find the area of the sector below ? www.mathsrevision.com Created by Mr Lafferty

31. Arc length is Pythagoras Theorem SOHCAHTOA Circumference is Area is Radius Diameter Sector area Summary of Circle Topic • line that bisects a chord • Splits the chord into 2 equal halves. • Makes right-angle with the chord. • Passes through centre of the circle Semi-circle angle is always 90o Tangent touches circle at one point and make angle 90o with point of contact radius www.mathsrevision.com Created by Mr Lafferty