![]() 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. 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. ![]() Lines in two dimensions Line forms Slope - intercept form: y mx b= + Two point form: ( )2 11 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. Download Analytic Geometry Formulas Cheat Sheet and more Analytical Geometry Cheat Sheet in PDF only on Docsity! Analytic Geometry Formulas 1.
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