Mathematics_Notes

Right-Angled Triangle Area of Triangle 1/2×𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡 Pythagoras’ Theorem 𝑎-+𝑏-=𝑐- Trigonometry TOA CAH SOH (Right-Angled Triangle) tan𝜃=𝑂𝐴 cos𝜃=𝐴𝐻 sin𝜃=𝑂𝐻 Non-Right-Angled Triangle (Given) Area of Triangle 12𝑎𝑏sin𝑐 Sine Rule <=>?@=<=>AB=<=>CD or vice versa Cosine Rule 𝑎-=𝑏-+𝑐-−2𝑏𝑐cos𝐴 Straight-Line Graphs Gradient of Line 𝑦-−𝑦G𝑥-−𝑥G Length of Line I(𝑥-−𝑥G)-+(𝑦-−𝑦G)- 𝑦=𝑚𝑥+𝑐 𝑚= gradient 𝑐=𝑦-intercept Acute, Obtuse and Reflex Angles Acute Angle 0°<𝜃<90° Obtuse Angle 90°<𝜃<180° Reflex Angle 180°<𝜃<360° Bearings Bearings always have up to hundreds place à 030° Standard Form Powers of 10 Name SI Prefix Symbol 10TG- Trillionth Pico- p 10TU Billionth Nano- n 10TV Millionth Micro- 𝜇 10TX Thousandth Milli- m 10Y One 10X Thousand Kilo- k 10V Million Mega- M 10U Billion Giga- G 10G- Trillion Tera- T Completing the Square 𝑎𝑥-+𝑏𝑥+𝑐 𝑎(𝑥-+𝑏𝑎𝑥+𝑐𝑎) 𝑎Z(𝑥+𝑏2𝑎)-−[𝑏2𝑎\-+𝑐𝑎] 𝑎(𝑥+ℎ)-+𝑘 𝑎(𝑥−ℎ)-+𝑘 Coefficient of 𝑥- must be +1 𝑎𝑥-−𝑏𝑥+𝑐 𝑎(𝑥-−𝑏𝑎𝑥+𝑐𝑎) 𝑎Z(𝑥−𝑏2𝑎)-−[𝑏2𝑎\-+𝑐𝑎] Turning point =(−ℎ,𝑘) Opposite (O) Hypotenuse (H) Adjacent (A) 𝜃

Interest Simple Interest 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙×𝑅𝑎𝑡𝑒100×𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑇𝑖𝑚𝑒𝑠 Compound Interest (Given) 𝑇𝑜𝑡𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙[1+𝑅𝑎𝑡𝑒100\mnoBpq rs tuopv Congruent Triangles Proof Explanation SSS (Side-Side-Side) All three corresponding sides are equal ASA (Angle-Side-Angle) Two corresponding angles and the line between them are equal SAS (Side-Angle-Side) Two corresponding sides and the angle between them are equal RHS (Right-angled Triangle, Hypotenuse, Side) Right-angled triangle, when hypotenuse and one corresponding side are equal Similar Triangles Proofs 1 Two corresponding angles are equal 2 Ratio of three corresponding sides are equal 3 Ratio of 2 corresponding sides and the angle between them are equal Similar Figures Length 1:5 Area 1:5-→1:25 Volume / Mass 1:5X→1:125 Arc Length and Sector Area Degree Radian (given) Arc Length 𝜃°360°×2𝜋𝑟 𝑟𝜃 Sector Area 𝜃°360°×𝜋𝑟- 12𝑟-𝜃 Conversion of Degrees to Radian 𝜋 𝑟𝑎𝑑=180° Degrees to Radian 𝜃°180°×𝜋 𝑟𝑎𝑑 Radian to Degrees 𝜃 𝑟𝑎𝑑𝜋 𝑟𝑎𝑑×180° Angle Properties Property Abbreviation Example Parallel Lines 𝑆𝑢𝑚 𝑜𝑓 𝑐𝑜−𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠=180° (co-int ∠s) 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 (alt ∠s) 𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 (Corresp ∠s) Polygon Sum of interior angles in a polygon =(𝑛−2)×180° Sum of exterior angles in a polygon =360° 𝜃 𝛼 𝜃 𝛼 𝜃 𝛼

Chords of Circle Perpendicular bisector of a chord of a circle passes through the centre of the circle (⊥ bisector of chord) Perpendicular line from centre of circle to a chord bisects the chord (⊥ from centre bisects chord) Equal chords of a circle are equidistant from the centre of the circle (⊥ Equal chords, equidistant from centre) Properties of Circle Property Short form Example Angle at centre = 2 x Angle at circumference (∠ at centre = 2 ∠ at circumference) Angle in a semicircle is a right angle (∠ in a semicircle) Angles in the same segment are equal (∠s in the same segment) Sum of angles in the opposite segments is 180° (∠s in opp. segments)