The Ungerecht Family Tree

delta x)² + (delta y)²) (based on Pythagorean theorem) length of a vector can be written as ||v|| (the "norm" of vector v) distance of two vectors: first add the vectors to make a combined vector adding vectors: add each part of the vector, ie. (x1 + x2, y1 + y2) ====Triangles==== centroid of a triangle - center (where the lines bisecting each angle will meet)

• average the x corner values, and average of y corner values
• the bisecting line will have 2/3 of its length between the corner and the centroid and 1/3 from centroid to far side of triangle
• the six small triangles formed by bisecting lines all have equal area

Heron's formula - area of any triangle, with sides of length a,b,c: semiperimeter (sp) = perimeter/2 = (a+b+c)/2 area = sqrt(sp(sp-a)(sp-b)(sp-c)) ===Right Triangles=== 45-45-90 triangle: hyp = side * sqrt(2) 30-60-90 triangle: short side = a, long side = a*sqrt(3), hyp = 2a area of a right triangle = 1/2*h*w law of sines: sin(A)/a = sin(B)/b = sin(C)/c (sometimes a/sin(A) = ...) law of cosines - to find an angle when all the sides are known cos(A) = (b^2 + c^2 - a^2) / (2bc) cos(C) = (a^2 + b^2 - c^2) / (2ab), cos(B) is same pattern (side a is opposite angle A, etc) rewriting it: c = sqrt(a^2 + b^2 - 2abcos(C)) ====Circles==== chords and circles : chord has two endpoints on a circle secant is a line that contains a chord, but extends beyond the circle if two chords AB, CD intersect at P, then AP * PB = CP * PD Intercepted arc = the part of the circle contained within the two lines Central angle = angle of two lines from the center of the circle Inscribed/Interior angle has two points and the vertex all on the circle itself central angle = same as the degrees of intercepted arc Interior angle = 1/2 of intercepted arc all inscribed angles going to the two same points on the circle have the same angle angle of intersecting secants theorem: angle formed by the secant intersection = (opposite arc - adjacent arc)/2 An angle outside the circle with two secants (or tangents) will have an angle that is 1/2 * (difference of the intercepted arcs) coterminal angle - the rest of the circle outside the angle. eg. angle of 30', the coterminal is 330' in radians, use the absolute value. 2pi - |angle|, or |angle| - 2pi for the negative angle area of an arc: a = rad*r^2/2 .... comes from a = pi*r^2 for full circle, and the proportion of a circle in the arc is rad/2pi so a = (rad/2pi)*(pi*r^2) ====Graphing==== vertex of a parabola: for y = ax^2 + bx + c, then x = -b/2a standard form of a parabola: (x-h)^2 = 4p(y-k) => if p>0, opens up, p<0 opens down (y-k)^2 = 4p(x-h) => if p>0, opens to right, p<0 opens to left where point (h,k) is the vertex, and p = minimum distance between parabola and vertex (is on axis of symmetry, which is perpendicular to the directrix) LR (latus rectum line) is line parallel to directrix going thru focus (if you know focus, easy to find LR points and vertex, then draw the function) length of LR is |4p| Standard form of a circle: (x-h)^2 + (y-k)^2 = r^2 where point (h,k) is the center, and r is radius This can be expanded to x^2 + y^2 + Dx + Ey + F = 0 (aka the General form) Can go from general form to standard form by completing the square Ellipse - sum of distance from two foci is a constant points on the ellipse follow the formula: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 and c^2 = a^2 - b^2 center C = (h,k) a = distance from C to long end of ellipse (along major axis) b = distance from C to close end of ellipse (along minor axis) c = distance from center to a focus (x,y) = a point on the ellipse Hyperbola - difference of distance from two foci is a constant === See Also ===

• https://www2.clarku.edu/faculty/djoyce/trig/identities.html
• 11 pages of definitions, postulates and theorems: http://www.ouchihs.org/ourpages/auto/2013/7/26/52822673/Geo-PostulatesTheorems-List.pdf

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