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Circles

Mathematics. Circles. Grade. INTRODUCTION. Objective : To identify some concepts of the circle such as center, chord, diameter and radius. P. CONTENTS. Circles. Diameter. Chord. Concepts on Circle. Quiz. Q. P. Circles.

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Circles

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  1. Mathematics Circles Grade

  2. INTRODUCTION Objective : To identify some concepts of the circle such as center, chord, diameter and radius.

  3. P CONTENTS Circles Diameter Chord Concepts on Circle Quiz Q

  4. P Circles A circle is the set of all points in a plane that are at a given distance from a given point. The given point P is called the center of the circle and the given distance is called theradiusof the circle. Radius Center Home

  5. P Diameter A diameter is a line segment that connects two points on the circle and goes through the centre of the circle. diameter A B

  6. P Central Angle A central angle is an angle whose vertex is the center of the circle.. A Central Angle B

  7. P Arc An arc is part of a circle connecting two points of the circle. A Arc B

  8. P Sector A sector is a region contained by two radii and an arc. A Sector B

  9. C D Chord A chord of a circle is a segment whose end points lie on the circle. chord

  10. C D P A B Q Concepts on circle Chord Central Angle Diameter Sector Arc Center Radius

  11. Quiz Choose the correct answer. 1. The longest chord of a circle is a A. diameter B. radius C. segment

  12. Quiz Choose the correct answer. 2. The distance between the center and a point on the circle is called A. diameter B. chord C. radius

  13. Quiz Choose the correct answer. 3. The part of a circle connecting two points of the circle is the A. central angle B. sector C. arc

  14. We use the Greek letter instead. The Circumference If it goes on for ever how can I write it down? Mathematical Genius! This is called pi.

  15. The Circumference So circumference ÷ diameter = 3.1415926535 By re-arranging this we get: x diameter Circumference = d C =

  16. When doing circle calculations, you will normally use a calculator. This button stores to 8 or 9 decimal places which is more than accurate enough! If your calculator doesn’t have Then use 3.14 instead. The Circumference Some calculators have a button like this: 3.141592654

  17. C = d C = ( 6) Press Then x 6 = Example 1 6cm C = 18.8cm What is the circumference of this circle?

  18. C = d C = x 10 Example 2 d = 2 x 5 = 10cm 10cm C = 31.4cm 5cm What is the circumference of this circle? Remember: diameter = 2 x radius

  19. ? ? ? ? ? ? 6 7 8 5 1 2 3 4 ? ? Area of a circle Mathematical Genius! To find the area we could try counting the squares inside the circle… There is a much more accurate way!

  20. x radius x radius Area = r² A= Area of a circle There is a special formula for the area of a circle. Remember: r² means r x r

  21. A = r² A = x 4 x 4 Press Then x 4 x4 = Example 1 4m A = 50.3m² What is the area of this circle?

  22. A = r² A = x 7 x 7 Press Then x 7 x 7 = Example 2 Don’t forget! r = ½ x 14 = 7cm 7cm ? A = 153.9cm² 14cm What is the area of this circle?

  23. A = r² A = x 12 x 12 24m Example 3 Don’t forget! r = ½ x 24 = 12m ? 12m A = 452.4m² What is the area of this semi-circle? Area of semi-circle = ½ x 452.4 =226.2m² A semicircle is half a circle. First work out area of full circle.

  24. Central Angles

  25. Central Angle : An Angle whose vertex is at the center of the circle ACB AB A Major Arc Minor Arc More than 180° Less than 180° P To name: use 3 letters C To name: use 2 letters B <APB is a Central Angle

  26. EDF Semicircle: An Arc that equals 180° To name: use 3 letters E D P F EF is a diameter, so every diameter divides the circle in half, which divides it into arcs of 180°

  27. THINGS TO KNOW AND REMEMBER ALWAYS A circle has 360 degrees A semicircle has 180 degrees Vertical Angles are Equal Linear Pairs are Supplementary

  28. Vertical Angles are Equal

  29. Linear Pairs are Supplementary http://www.mathopenref.com/linearpair.html 120° 60°

  30. measure of an arc = measure of central angle m AB m ACB m AE A E 96 Q = 96° B C = 264° = 84°

  31. Arc Addition Postulate m ABC = m AB + m BC A C B

  32. m DAB = Tell me the measure of the following arcs. 240 D A 140 260 m BCA = R 40 100 80 C B

  33. CONGRUENT ARCS Congruent Arcs have the same measure and MUST come from the same circle or from congruent circles. C B D 45 45 110 A

  34. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle INTERCEPTEDARC INSCRIBEDANGLE

  35. YES; CL C T O L Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 1.

  36. NO; QVR Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 2. Q V K R S

  37. To find the measure of an inscribed angle… 160° 80°

  38. What do we call this type of angle? What is the value of x? What do we call this type of angle? How do we solve for y? The measure of the inscribed angle is HALF the measure of the inscribed arc!! 120 x y

  39. J K Q S M Examples 3. If m JK = 80, find m<JMK. 40  4. If m<MKS = 56, find m MS. 112 

  40. If two inscribed angles intercept the same arc, then they are congruent. 72

  41. Q D A J T B U Example 5 In J, m<A= 5x and m<B = 2x + 9. Find the value of x. m<A = m<B 5x = 2x+9 x = 3

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