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S5_Int2_The_Circle

# S5_Int2_The_Circle - The Circle Int 2 www.mathsrevision.com...

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Unformatted text preview: The Circle Int 2 www.mathsrevision.com Length of Arc in a Circles Area of a Sector in a Circle Mix Problems including Angles 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 Monday 13 February 2012 Created by Mr Lafferty 1 Starter Questions Int 2 www.mathsrevision.com Q1. Simplify 5( x + 2) + 2 x(a + 3) Q2. How many degrees in one eighth of a circle. Q3. Fac t o r is e x 2 - 8 x + 12 Q4. After a discount of 20% an iPod is 160. How much was it originally. Monday 13 February 2012 Created by Mr Lafferty 2 Arc length of a circle Int 2 www.mathsrevision.com Aim of Today's Lesson To find and be able to use the formula for calculating the length of an arc. Monday 13 February 2012 Created by Mr Lafferty 3 Arc length of a circle Int 2 Q. What is an arc ? www.mathsrevision.com A Answer An arc is a fraction of the circumference. minor arc B major arc Monday 13 February 2012 Created by Mr Lafferty 4 Arc length of a circle Int 2 Q. Find the circumference of the circle ? Solution 10cm www.mathsrevision.com C = D C = 3 .14 2 0 C = 6 2 .8 c m Monday 13 February 2012 Created by Mr Lafferty 5 Arc length of a circle Int 2 Q. Find the length of the minor arc XY below ? www.mathsrevision.com x y 45 o connection Arc length D 6 cm Arc angle = 360o 360 o 45o ar c le ng t h = 360o (3 .14 12 ) ar c le ng t h = 4 .7 1c m Monday 13 February 2012 Created by Mr Lafferty 6 Arc length of a circle Int 2 Q. Find the length of the minor arc AB below ? www.mathsrevision.com A 9 cm 60 o connection Arc length D Arc angle = 360o 60o ar c le ng t h = o 360 B (3 .14 18 ) ar c le ng t h = 9 .4 2 c m Created by Mr Lafferty 7 Monday 13 February 2012 Arc length of a circle Int 2 Q. Find the length of the major arc PQ below ? www.mathsrevision.com A 10 m 260 o o connection Arc length D Arc angle = 360o 100 260o ar c le ng t h = o 360 B (3 .14 20) ar c le ng t h = 4 5 .3 8 c m Created by Mr Lafferty 8 Monday 13 February 2012 Arc length of a circle Int 2 www.mathsrevision.com Now try Ex 1 Ch6 MIA (page 60) 13 Feb 2012 Created by Mr. [email protected] Starter Questions Int 2 www.mathsrevision.com Q1. Solve 5( x + 2) = 20 Q2. How many degrees in one tenth of a circle. Q3. Fac t o r is e x 2 - 13 x + 4 2 Q4. After a discount of 40% a Digital Radio is 120. How much was it originally. Monday 13 February 2012 Created by Mr Lafferty 10 Sector area of a circle Int 2 www.mathsrevision.com Aim of Today's Lesson To find and be able to use the formula for calculating the sector of an circle. Monday 13 February 2012 Created by Mr Lafferty 11 Area of Sector in a circle Int 2 www.mathsrevision.com A minor sector B major sector Monday 13 February 2012 Created by Mr Lafferty 12 Area of Sector in a circle Int 2 Q. Find the area of the circle ? Solution 10cm www.mathsrevision.com A = r 2 A = 3 .14 10 2 A = 3 14 c m 2 Monday 13 February 2012 Created by Mr Lafferty 13 Area of Sector in a circle Int 2 Find the area of the minor sector XY below ? www.mathsrevision.com x y 45 o connection 6 cm Area Sector r2 = Sector angle 360o (3 .14 62) 360 o 45o A r e a o f S e c t o r = 360o A r e a S e c t o r = 14 .14 c m 2 Monday 13 February 2012 Created by Mr Lafferty 14 Area of Sector in a circle Int 2 Q. Find the area of the minor sector AB below ? www.mathsrevision.com connection A 9 cm 60 o Area Sector r2 = Sector angle 360o 60o A r e a S e c t o r = 360o B (3 .14 9 ) 2 A r e a S e c t o r = 4 2 .4 1c m 2 Created by Mr Lafferty 15 Monday 13 February 2012 Area of Sector in a circle Int 2 Q. Find the area of the major sector PQ below ? www.mathsrevision.com connection A Sector Area r2 10 m 260 o Sector angle = 360o 100 o 260o S e c t o r A r e a = 360o B (3 .14 10 2 ) A r e a S e c t o r = 2 2 6 .8 9 c m 2 Created by Mr Lafferty 16 Monday 13 February 2012 Area of Sector in a circle Int 2 www.mathsrevision.com Now try Ex 2 Ch6 MIA (page 62) 13 Feb 2012 Created by Mr. [email protected] Starter Questions Int 2 www.mathsrevision.com Q1. Find the gradient and y intercept 2 y = 4 x + 16 2 Q2. Expand out (x2 + 4x 3)(x + 1) Q3. Fac t o r is e 4 x -9 Q4. I want to make 15% profit on a computer I bought for 980. How much must I sell it for. Monday 13 February 2012 Created by Mr Lafferty 18 Arc length of a circle Int 2 Q. The arc length is 7.07cm. Find the angle xo www.mathsrevision.com A 9 cm connection Arc length D Arc angle = 360o x o x B o 7 .0 7 = (3 .14 18 ) x o 360o = 45o 19 Monday 13 February 2012 Created by Mr Lafferty Area of Sector in a circle Int 2 Q. Find the angle given area of sector AB is 235.62cm2 ? www.mathsrevision.com connection A Sector Area r2 o 10 m Sector angle = 360o x o 2 3 5 .6 2 x = (3 .14 10 2 ) B 360o xo = 270o Created by Mr Lafferty 20 Monday 13 February 2012 Mixed Problems Int 2 www.mathsrevision.com Now try Ex 3 Ch6 MIA (page 64) 13 Feb 2012 Created by Mr. [email protected] Starter Questions Int 2 www.mathsrevision.com Q1. Find the gradient and y intercept -2 x + 2 y = 20 2 Q2. Expand out (x + 3)(x2 + 40 9) Q3. Fac t o r is e x - 4x + 4 Q4. I want to make 30% profit on a DVD player I bought for 80. How much must I sell it for. Monday 13 February 2012 Created by Mr Lafferty 22 Int 2 Isosceles triangles in Circles Aim of Today's Lesson To identify isosceles triangles within a circle. www.mathsrevision.com Monday 13 February 2012 Created by Mr Lafferty 23 Int 2 Isosceles triangles in Circles When two radii are drawn to the ends of a chord, An isosceles triangle is formed. A Online Demo B www.mathsrevision.com xo x o C Monday 13 February 2012 Created by Mr Lafferty 24 Int 2 Isosceles triangles in Circles Special Properties of Isosceles Triangles www.mathsrevision.com Two equal lengths Two equal angles Angles in any triangle sum to 180 Monday 13 February 2012 Created by Mr Lafferty o 25 Int 2 Isosceles triangles in Circles Q. Find the angle x . o www.mathsrevision.com Solution Angle at C is equal to: 360o -280o = 80o B A x o Since the triangle is isosceles we have 2x 2x x Monday 13 February 2012 o o + 8 0 o = 18 0 o o 280 o C 18 0 o - 8 0 o = 2 x o = 10 0 o = 50o Created by Mr Lafferty 26 Int 2 Isosceles triangles in Circles www.mathsrevision.com Now try Ex 4 Ch6 MIA (page 66) 13 Feb 2012 Created by Mr. [email protected] Starter Questions Int 2 www.mathsrevision.com Q1. Find the gradient and y intercept 1 x = 2 y + 10 2 2 Q2. Expand out 2(x 3) + 3x Q3. Fac t o r is e 3 x + x -2 Q4. Car depreciates by 20% each year. How much is it worth after 3 years. Monday 13 February 2012 Created by Mr Lafferty 28 Diameter symmetry Int 2 www.mathsrevision.com Aim of Today's Lesson To understand some special properties when a diameter bisects a chord. Monday 13 February 2012 Created by Mr Lafferty 29 Diameter symmetry Int 2 C o A D B 1. A line drawn through the centre of a circle and through the midpoint a chord will ALWAYS cut the chord at rightangles 2. A line drawn through the centre of a circle at rightangles 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. Created by Mr Lafferty 30 Monday 13 February 2012 Diameter symmetry Int 2 Q. Find the length of the chord A and B. www.mathsrevision.com Solution Radius of the circle is 4 + 6 = 10. Since yellow line bisect AB and passes through centre O, triangle is rightangle. By Pythagoras Theorem we have a +b 2 2 2 2 B 10 4 6 O =c 2 2 a + 6 = 10 a 2 = 10 2 - 6 2 a 2 = 10 0 - 3 6 = 6 4 a = 64 = 8 Monday 13 February 2012 Since AB is bisected The length of AB is le ng t hA B = 2 8 = 16 A Created by Mr Lafferty 31 Diameter symmetry Int 2 Find the length of OM and CM. Radius of circle is 5cm, AB is 8cm and M is midpoint of AB. C Radius of the circle is 5cm. o A M D Monday 13 February 2012 AM is 8 2 = 4 By Pythagoras Theorem we have B O M 2 + AM 2 = O A 2 O M2 + 42 = 52 O M2 = 52 - 42 O M 2 = 2 5 - 16 = 9 O M = 9 = 3c m Created by Mr Lafferty CM = 3cm + radius CM = 3 + 5 = 8cm 32 Int 2 Symmetry & Chords in Circles www.mathsrevision.com Now try Ex 5 Ch6 MIA (page 68) 13 Feb 2012 Created by Mr. [email protected] Starter Questions Int 2 www.mathsrevision.com Q1. Find the gradient and y intercept 5x + y + 1 = 0 Q2. Expand out (x + 3)(x + 4) Q3. Fac t o r is e 2 d 2 g + d g Q4. Bacteria increase at a rate of 10% per hour. If there were 100 bacteria initial. How many are there after 9 hours. Monday 13 February 2012 Created by Mr Lafferty 34 Semi-Circle Angle Int 2 www.mathsrevision.com Aim of Today's Lesson To find the angle in a semicircle made by a triangle with hypotenuse equal to the diameter and the two smaller lengths meeting at the circumference. Monday 13 February 2012 Created by Mr Lafferty 35 Semi-circle angle Int 2 Toolkit required www.mathsrevision.com 1. Protractor 2. Pencil 3. Ruler Monday 13 February 2012 Created by Mr Lafferty 36 Semi-circle angle Int 2 www.mathsrevision.com 1. Using your pencil trace round the protractor so that you have semicircle. 2. Mark the centre of the semicircle. You should have something like this. Monday 13 February 2012 Created by Mr Lafferty 37 Semi-circle angle Int 2 Mark three points x x x x www.mathsrevision.com x x x 1. Outside the circle 2. On the circumference 3. Inside the circle x x Monday 13 February 2012 Created by Mr Lafferty 38 Semi-circle angle Int 2 For each of the points x www.mathsrevision.com x x Form a triangle by drawing a line from each end of the diameter to the point. Measure the angle at the various points. Outside Log your results in a table. Circumference Inside Monday 13 February 2012 Created by Mr Lafferty 39 Semi-circle angle Int 2 www.mathsrevision.com Online Demo Outside Circumference x x Inside x < 90 o = 90 o > 90 o Begin Maths in Action Book page 182 Monday 13 February 2012 Created by Mr Lafferty 40 Angles in a Semi-Circle Int 2 www.mathsrevision.com Now try Ex 7 Ch6 MIA (page 71) 13 Feb 2012 Created by Mr. [email protected] Starter Questions Int 2 www.mathsrevision.com Q1. Find the gradient and y intercept -8 x + 2 y = 16 2 Q2. Expand out (a 3)(a2 + 3a 7) Q3. Fac t o r is e x -36 Q4. I want to make 60% profit on a TV I bought for 240. How much must I sell it for. Monday 13 February 2012 Created by Mr Lafferty 42 Tangent line Int 2 www.mathsrevision.com Aim of Today's Lesson To understand what a tangent line is and its special property with the radius at the point of contact. Monday 13 February 2012 Created by Mr Lafferty 43 Tangent line Int 2 www.mathsrevision.com A tangent line is a line that touches a circle at only one point. Which of the lines are tangent to the circle? Monday 13 February 2012 Created by Mr Lafferty 44 Tangent line Int 2 www.mathsrevision.com The radius of the circle that touches the tangent line is called the point of contact radius. Online Demo Special Property The point of contact radius is always perpendicular (rightangled) to the tangent line. Monday 13 February 2012 Created by Mr Lafferty 45 Tangent line Int 2 www.mathsrevision.com Solution Rightangled at A since AC is the radius at the point of contact with the Tangent. a2 +b 2 =c 2 a + 8 = 10 2 2 2 Q. Find the length of the tangent line between A and B. B 10 8 C By Pythagoras Theorem we have A a 2 = 10 2 - 8 2 a 2 = 10 0 - 6 4 = 3 6 a = 36 = 6 Monday 13 February 2012 Created by Mr Lafferty 46 Tangent Lines Int 2 www.mathsrevision.com Now try Ex 8 Ch6 MIA (page 73) 13 Feb 2012 Created by Mr. [email protected] Summary of Circle Topic Int 2 Arc length is A r c le ng t h = c e nt r e a ng le o 360o D 1. 2. line that bisects a chord www.mathsrevision.com 3. Circumference is Splits the chord into 2 equal halves. Makes rightangle with the chord. Passes through centre of the circle Pythagoras Theorem SOHCAHTOA C = D Tangent touches circle at one point o and make angle 90 with point of contact radius Semicircle angle is o always 90 A = r Radius r = 1 D 2 Sector area c e nt r e ang le o S e c t o r = 360o Area is 2 D = 2r Monday 13 February 2012 Diameter Created by Mr Lafferty r 2 48 ...
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