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Algebra: Co-ordinate Geometry: Properties of Circles:  𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐  (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2...

Algebra: Co-ordinate Geometry: Properties of Circles:  𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐  (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 Angle at  (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 centre (2p) is  𝑎2 + 𝑏 2 = (𝑎 + 𝑏)2 − 2𝑎𝑏 twice angle at  𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) circumference (p) Quadratic Equations: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 −𝑏±√𝑏 2 −4𝑎𝑐 Solve by formula: 𝑥 = 2𝑎 -------------------------------------------- Complete the square: Eqn. of a straight line: 𝑦 = 𝑚𝑥 + 𝑐 Make sure a = 1 𝑦2 − 𝑦1 Angle in the same Gradient of a straight line: 𝑚 = 𝑏 𝑏 𝑥2 − 𝑥1 segment of a circle are 𝑎𝑥 2 + 𝑏𝑥 + ( )2 − ( )2 + 𝑐 = 0 𝑥1 + 𝑥2 𝑦1 + 𝑦2 2 2 Midpoint: 𝑀 = ( , ) equal 2 2 Indices: Distance between two points: --------------------------------------------  𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛 𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 Angle in a semicircle  𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛 Matrices: is a right angle.  (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 Addition  𝑎0 = 1 𝑎 𝑏 𝑝 𝑞 𝑎+𝑝 𝑏+𝑞 1 ( )+( )=( ) --------------------------------------------  𝑎 −𝑛 = 𝑛 𝑐 𝑑 𝑟 𝑠 𝑐+𝑟 𝑑+𝑠 𝑎 Opposite angles of a  (𝑎 × 𝑏)𝑚 = 𝑎𝑚 × 𝑏 𝑚 Subtraction quadrilateral add up 𝑎 𝑎𝑚 𝑎 𝑏 𝑝 𝑞 𝑎−𝑝 𝑏−𝑞  ( )𝑚 = ( )−( )=( ) to 180o 𝑏 𝑏𝑚 𝑐 𝑑 𝑟 𝑠 𝑐−𝑟 𝑑−𝑠 𝑚 ∠A + ∠C = 180o  𝑛 ( √𝑎 )𝑚 = 𝑎 𝑛 Multiplication ∠B + ∠D = 180o  √𝑎 × √𝑏 = √𝑎 × 𝑏 𝑎 𝑏 𝑝 𝑞 𝑎𝑝 + 𝑏𝑟 𝑎𝑞 + 𝑏𝑠 -------------------------------------------- ( )×( )=( ) 𝑎 √𝑎 𝑐 𝑑 𝑟 𝑠 𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠  √𝑏 = √𝑏 Exterior angle of a 𝑎 𝑏 𝑘𝑎 𝑘𝑏 quadrilateral equals 𝑘×( )=( )  (√𝑎)2 = 𝑎 𝑐 𝑑 𝑘𝑐 𝑘𝑑 to interior opposite Variation: Vectors: angle (∠b = ∠p) 𝑦 is proportional to 𝑥: 𝑦 = 𝑘𝑥 Triangular law 𝑘 of addition: -------------------------------------------- 𝑦 is inversely proportional to 𝑥: 𝑦 = 𝑥 Chord of a Circle: 𝑃𝑅𝑇 Simple Interest - To find interest: 𝑖 = ⃗⃗⃗⃗⃗ + 𝐴𝐶 𝑂𝐴 ⃗⃗⃗⃗⃗ = 𝑂𝐶 ⃗⃗⃗⃗⃗ 100 𝑟 A line joining two Compound Interest: 𝐴 = 𝑃(1 + )𝑛 Parallelogram law of addition: 100 points on a circle is ⃗⃗⃗⃗⃗ + 𝑂𝐴 𝑂𝐵 ⃗⃗⃗⃗⃗ = 𝑂𝐶 ⃗⃗⃗⃗⃗ Conversion of units: called a chord (Line AB). 5 18 km/hr x = m/sec ; m/sec x = km/hr Polygons: 18 5 Sum of Exterior angles = 360°  𝟐 𝟐 𝟐 Pythagoras Theorem: 𝑐 = 𝑎 + 𝑏 -------------------------------------------- Trigonometry: One Exterior angle = 360° Tangent to a Circle: 𝑛 TOA CAH SOH Angle between Sum of interior angles = (𝑛 − 2) × 180° tangent and radius drawn to contact Types of polygons: ∠ABO or ∠OBC = 90° 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑂 No. of sides 𝑇𝑎𝑛 𝜃 = = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴 -------------------------------------------- 4 Quadrilateral 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴 5 Pentagon Any point outside of a 𝐶𝑜𝑠 𝜃 = = 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 𝐻 6 Hexagon circle, two tangents 7 Heptagon drawn to the circle = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑂 𝑆𝑖𝑛 𝜃 = = 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 𝐻 8 Octagon equal length (TA = TB) 9 Nonagon 10 decagon -------------------------------------------- Alternate Segment Theorem: Sine Rule: Angle Properties of Triangle: Cosine Rule: 𝑎 𝑏 𝑐 Length of a side: 𝑎2 = 𝑏 2 + 𝑐 2=− 2𝑏𝑐=cos 𝐴 ∠QAB = ∠ACB (p = q) sin 𝐴 sin 𝐵 sin 𝐶 Find an angle when all 3 sides are given: 𝑏2 + 𝑐 2 − 𝑎2 cos 𝐴 = 2𝑏𝑐 Sum of all angles = 180° TC2 = AC x BC Exterior angle (x) = Sum of opposite interior angles (𝑎 + 𝑏) ∠QAB = ∠ACB (p = q) Similar Plane Figures Congruent Figures Graphs of functions - Figures are similar only if Congruent figures are exactly the same Positive Negative their corresponding sides are size and shape. proportional y = mx + c y = -mx + c - their corresponding angles are 2 triangles are congruent if they satisfy equal any of the following: a. SSS property: All 3 sides of one triangle are equal to the corresponding sides of the other triangle. 𝒚 = 𝑎𝑥 𝟐 + 𝒃𝒙 + 𝒄 𝒚 = −𝑎𝑥 𝟐 + 𝒃𝒙 + 𝒄 b. SAS property: 2 given sides and a 𝒚 = 𝑎𝑥 𝟑 𝒚 = −𝑎𝑥 𝟑 given angle of one triangle are equal to the corresponding sides and angle of the other triangle. 𝒚 = 𝑎𝑥 −𝟏 𝒚 = −𝑎𝑥 −𝟏 c. AAS property: 2 given angles and a given side of one triangle are equal to the corresponding angles and side of the other Similar Solid Figures triangle. 𝒚 = 𝑎𝑥 −𝟐 𝒚 = −𝑎𝑥 −𝟐 Solids are similar if their corresponding linear dimensions are proportional. 𝒚 = 𝒌𝑎𝒙 𝒚 = −𝑘𝑎𝒙 d. RHS property: The a>1 a>1 hypothenuse and a given side of a right-angled triangle are equal to the hypothenuse and the corresponding side of the other right-angled triangle. Graphs from complete the square 𝒚 = (𝑥 + 𝑎)𝟐 + 𝒉 𝒚 = −(𝑥 + 𝑎)𝟐 + 𝒉 min 𝑝𝑜𝑖𝑛𝑡 (−𝑎, ℎ) max 𝑝𝑜𝑖𝑛𝑡 (−𝑎, ℎ) Area & Perimeter: Surface Area & Volume: Figure Area Perimeter/ Figure Surface area Volume Circumference Cylinder Rectangle Curved surface area = 2𝜋𝑟ℎ 𝜋𝑟 2 ℎ 𝑙×𝑏 2 (𝑙 + 𝑏) Total surface area = 2𝜋𝑟(ℎ + 𝑟) Square Cone Curved surface area = 𝜋𝑟𝑙 𝑎×𝑎 4×𝑎 1 2 Where 𝑙 = √(𝑟 2 + ℎ2 ) 𝜋𝑟 3 Total surface area = 𝜋𝑟(𝑙 + 𝑟) Parallelogram Sphere 𝑏×ℎ 2(𝑎 + 𝑏) 4 3 4𝜋𝑟 2 𝜋𝑟 3 Triangle 1 ×𝑏×ℎ Pyramid 2 Or 𝑎+𝑏+𝑐 1 × base area × 1 3 𝑎𝑏 sin 𝐶 Base area + Area of shapes in the sides perpendicular 2 height Trapezium 1 Cubiod (𝑎 + 𝑏)ℎ Sum of all sides 2 2(𝑙𝑏 + 𝑏ℎ + 𝑙ℎ) 𝑙×𝑏×ℎ Circle 𝜋𝑟 2 2𝜋𝑟 Cube 6𝑙2 𝑙3 Semicircle 1 2 1 𝜋𝑟 𝜋𝑑 + 𝑑 2 2 Hemisphere 2 3 Sector 2𝜋𝑟 2 𝜋𝑟 Length of an arc = 3 𝜃 𝜋𝑟 2 × 𝜃 360 2𝜋𝑟 × 360 Sets: Statistics: Probability: ∑ 𝑓𝑥 𝑛𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Mean = = average Prob. Of an event = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 ∑𝑓 Mode of a series of number = number which occurs Exclusive event (events cannot occur at the same time) most frequently For exclusive event A & B: p(A or B) = p(A) + p(B) Median = arrange series of numbers in ascending Independent event (events can occur at the same time) order and then choosing the number in the middle. For independent event A & B: p(A and B) = p(A) × p(B) Ace AceScorers ScorersEnrichment EnrichmentCentre Centre2A Maju Avenue S556680 Contact Contactusus@@6289 62894551 4551oror6440 64400553 0553 Youtube Ace Scorers www.acescorers.com.sg www.acescorers.com.sg Instagram @chernteng