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## The Circle

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**The Circle**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**Starter Questions**Q1. Simplify 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.**Arc length of a circle**Aim of Today’s Lesson To find and be able to use the formula for calculating the length of an arc.**minor arc**major arc Q. What is an arc ? Arc length of a circle Answer An arc is a fraction of the circumference. A B**Solution**Q. Find the circumference of the circle ? Arc length of a circle 10cm**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 360o**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 60o B**Q. Find the length of the major arc PQ below ?**Arc length of a circle Arc length Arc angle = connection A πD 360o 10 m 260o 100o B**Arc length of a circle**Now try Ex 1 Ch6 MIA (page 60)**Starter Questions**Q1. Solve Q2. How many degrees in one tenth of a circle. Q3. Q4. After a discount of 40% a Digital Radio is £120. How much was it originally.**Sector area of a circle**Aim of Today’s Lesson To find and be able to use the formula for calculating the sector of an circle.**minor sector**major sector Area of Sector in a circle A B**Solution**Q. Find the area of the circle ? Area of Sector in a circle 10cm**Find the area of the minor sector XY below ?**Area of Sector in a circle x connection Area Sector Sector angle = πr2 y 360o 6 cm 45o 360o**Q. Find the area of the minor sector AB below ?**Area of Sector in a circle connection Area Sector Sector angle = A πr2 360o 9 cm 60o B**Q. Find the area of the major sector PQ below ?**Area of Sector in a circle connection Sector Area Sector angle = A πr2 360o 10 m 260o 100o B**Area of Sector in a circle**Now try Ex 2 Ch6 MIA (page 62)**Starter Questions**Q1. Find the gradient and y - intercept Q2. Expand out (x2 + 4x - 3)(x + 1) Q3. Q4. I want to make 15% profit on a computer I bought for £980. How much must I sell it for.**Q. The arc length is 7.07cm. Find the angle xo**Arc length of a circle Arc length Arc angle = connection A πD 360o 9 cm xo B**Q. Find the angle given area of sector AB is 235.62cm2 ?**Area of Sector in a circle connection Sector Area Sector angle = A πr2 360o 10 m xo B**Mixed Problems**Now try Ex 3 Ch6 MIA (page 64)**Starter Questions**Q1. Find the gradient and y - intercept Q2. Expand out (x + 3)(x2 + 40 – 9) Q3. Q4. I want to make 30% profit on a DVD player I bought for £80. How much must I sell it for.**Isosceles triangles in Circles**Aim of Today’s Lesson To identify isosceles triangles within a circle.**A**B Isosceles triangles in Circles When two radii are drawn to the ends of a chord, An isosceles triangle is formed. xo xo Online Demo C**Isosceles triangles in Circles**Special Properties of Isosceles Triangles Two equal lengths Two equal angles Angles in any triangle sum to 180o**Solution**Angle at C is equal to: Isosceles triangles in Circles Q. Find the angle xo. B C xo Since the triangle is isosceles we have A 280o**Isosceles triangles in Circles**Now try Ex 4 Ch6 MIA (page 66)**Starter Questions**Q1. Find the gradient and y - intercept Q2. Expand out 2(x - 3) + 3x Q3. Q4. Car depreciates by 20% each year. How much is it worth after 3 years.**Diameter symmetry**Aim of Today’s Lesson To understand some special properties when a diameter bisects a chord.**C**A B D Diameter symmetry • A line drawn through the centre of a circle • and 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 (cut equally in 2) that chord. • A line bisecting a chord at right angles • will ALWAYS pass through the centre of a circle.**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 By Pythagoras Theorem we have 4 6 Since AB is bisected The length of AB is A**A**B Diameter symmetry 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. AM is 8 ÷ 2 = 4 o By Pythagoras Theorem we have M CM = 3cm + radius D CM = 3 + 5 = 8cm**Symmetry & Chords in Circles**Now try Ex 14.2 Ch14 (page 188)**Starter Questions**Q1. Find the gradient and y - intercept Q2. Expand out (x + 3)(x + 4) Q3. Q4. Bacteria increase at a rate of 10% per hour. If there were 100 bacteria initially. How many are there after 9 hours.**Semi-Circle Angle**Aim of Today’s Lesson 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.**Semi-circle angle**Tool-kit required 1. Protractor 2. Pencil 3. Ruler**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.**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**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**Outside**Circumference Inside Semi-circle angle x Online Demo x x = 90o > 90o < 90o Every angle in a semi-circle is a right angle**Angles in a Semi-Circle**Now try Ex 7 Ch6 MIA (page 71)**Starter Questions**Q1. Find the gradient and y - intercept Q2. Expand out (a - 3)(a2 + 3a – 7) Q3. Q4. I want to make 60% profit on a TV I bought for £240. How much must I sell it for.**Tangent line**Aim of Today’s Lesson To understand what a tangent line is and its special property with the radius at the point of contact.**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.**Tangent line**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 (right-angled) to the tangent line.**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 By Pythagoras Theorem we have C A 8**Tangent Lines**Now try Ex 8 Ch6 MIA (page 73)**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