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--- urp:geometry [2021-10-13]
nerf_herder [Angles]
+++ urp:geometry [2021-11-17] (current)
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 =====Geometry Reference Page=====
-Contents:
+  * [[Angles|Angles and Vectors]]
-  * [[#Angles|Angles and Polygons]]
+  * [[Triangles]]
-  * [[#Triangles|Right triangles]]
+  * [[Circles|Circles, Chords, etc]]
-  * [[#Circles|Circles and Chords]]
+  * [[Polygons]]
-  * [[#Graphing|Graphing curves, circles, etc.]]
+=== misc ===
+Truth tables:
+first columns : True/False values of the variables, such as p, q, or p' or ~p for inverse values. Next columns are logic combinations of the variables
-====Angles====
+| p | q | p ⋂ q | p ⋃ q| p => q |
+| T | T | T | T | T |
-Complementary angles add up to 90' (like the two non-right angles in a right triangle)
+| T | F | F | T | F |
+| F | T | F | T | T |
-Supplementary angles add up to 180'
+| F | F | F | F | T |
-The sum of angles in a polygon with n sides:  = 180 (n - 2).  Can also be written as: 180 + 180 (n-3).
-(Basically you add in another triangle when adding a side to polygon)
-distance between two points: √((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
-<nowiki>
-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))
-</nowiki>
-===Right Triangles===
--45-90 triangle: hyp = side * sqrt(2)
--60-90 triangle: short side = a, long side = a*sqrt(3), hyp = 2a
-area of a right triangle = 1/2*h*w
-<nowiki>
-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))
-</nowiki>
-====Circles====
-<nowiki>
-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.
-pi - |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)
-</nowiki>
-====Graphing====
-<nowiki>
-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
+p => q means condition p implies condition q. If p is true, then an implication can be drawn or not, depending on q. If p is false, then an implication cannot be ruled out, regardless of q, so it is left as true.
-  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
-</nowiki>
 === 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
 Back to [[math]] page.

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