Analytical Geometry Formula Sheet, Cheat Sheet of Mathematics

Download Analytical Geometry Formula Sheet and more Cheat Sheet Mathematics in PDF only on Docsity! Analytic Geometry Formulas 1. Lines in two dimensions Line forms Slope - intercept form: y mx b= + Two point form: ( )2 1 1 1 2 1 y y y y x x x x − − = − − Point slope form: ( )1 1y y m x x− = − Intercept form ( )1 , 0 x y a b a b + = ≠ Normal form: cos sinx y pσ σ⋅ + = Parametric form: 1 1 cos sin x x t y y t α α = + = + Point direction form: 1 1 x x y y A B − − = where (A,B) is the direction of the line and 1 1 1 ( , )P x y lies on the line. General form: 0 0 0A x B y C A or B⋅ + ⋅ + = ≠ ≠ Distance The distance from 0Ax By C+ + = to 1 1 1 ( , )P x y is 1 1 2 2 A x B y C d A B ⋅ + ⋅ + = + Concurrent lines Three lines 1 1 1 2 2 2 3 3 3 0 0 0 A x B y C A x B y C A x B y C + + = + + = + + = are concurrent if and only if: 1 1 1 2 2 2 3 3 3 0 A B C A B C A B C = Line segment A line segment 1 2 PP can be represented in parametric form by ( ) ( ) 1 2 1 1 2 1 0 1 x x x x t y y y y t t = + − = + − ≤ ≤ Two line segments 1 2PP and 3 4P P intersect if any only if the numbers 2 1 2 1 3 1 3 1 3 1 3 1 3 4 3 4 2 1 2 1 2 1 2 1 3 4 3 4 3 4 3 4 x x y y x x y y x x y y x x y y s and t x x y y x x y y x x y y x x y y − − − − − − − − = = − − − − − − − − satisfy 0 1 0 1s and t≤ ≤ ≤ ≤ 2. Triangles in two dimensions Area The area of the triangle formed by the three lines: 1 1 1 2 2 2 3 3 3 0 0 0 A x B y C A x B y C A x B y C + + = + + = + + = is given by 2 1 1 1 2 2 2 3 3 3 2 21 1 3 3 3 32 2 1 1 2 A B C A B C A B C K A BA B A B A BA B A B = ⋅ ⋅ ⋅ The area of a triangle whose vertices are 1 1 1( , )P x y , 2 2 2 ( , )P x y and 3 3 3 ( , )P x y : 1 1 2 2 3 3 1 1 1 2 1 x y K x y x y = 2 1 2 1 3 1 3 1 1 . 2 x x y y K x x y y − − = − − Centroid The centroid of a triangle whose vertices are 1 1 1 ( , )P x y , 2 2 2( , )P x y and 3 3 3( , )P x y : 1 2 3 1 2 3( , ) , 3 3 x x x y y y x y + + + +  =     Incenter The incenter of a triangle whose vertices are 1 1 1 ( , )P x y , 2 2 2( , )P x y and 3 3 3( , )P x y : 1 2 3 1 2 3( , ) , ax bx cx ay by cy x y a b c a b c + + + +  =   + + + +  where a is the length of 2 3 P P , b is the length of 1 3 PP , and c is the length of 1 2.PP Circumcenter The circumcenter of a triangle whose vertices are 1 1 1 ( , )P x y , 2 2 2 ( , )P x y and 3 3 3 ( , )P x y : 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 1 1 ( , ) , 1 1 2 1 2 1 1 1 x y y x x y x y y x x y x y y x x y x y x y x y x y x y x y x y  + +   + +   + +  =           Orthocenter The orthocenter of a triangle whose vertices are 1 1 1 ( , )P x y , 2 2 2 ( , )P x y and 3 3 3 ( , )P x y : 2 2 1 2 3 1 1 2 3 1 2 2 2 3 1 2 2 3 1 2 2 2 3 1 2 3 3 1 2 3 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 1 1 ( , ) , 1 1 2 1 2 1 1 1 y x x y x y y x y x x y x y y x y x x y x y y x x y x y x y x y x y x y x y  + +   + +   + +  =           3. Circle Equation of a circle In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that: ( ) ( ) 2 2 2 x a y b r− + − = Circle is centred at the origin 2 2 2 x y r+ = Parametric equations cos sin x a r t y b r t = + = + where t is a parametric variable. In polar coordinates the equation of a circle is: ( )2 2 22 coso or rr r aθ ϕ− − + = Area 2 A r π= Circumference 2c d rπ π= ⋅ = ⋅ Theoremes: (Chord theorem) The chord theorem states that if two chords, CD and EF, intersect at G, then: CD DG EG FG⋅ = ⋅ (Tangent-secant theorem) If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then 2 DC DG DE= ⋅ (Secant - secant theorem) If two secants, DG and DE, also cut the circle at H and F respectively, then: DH DG DF DE⋅ = ⋅ (Tangent chord property) The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord.