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--- urp:geometry [2021-10-13]
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+++ urp:geometry [2021-10-21]
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   * [[Angles|Angles and Vectors]]
-  * [[Polygons]]
   * [[Triangles]]
   * [[Circles|Circles, Chords, etc]]
-  * [[Graphing|Graphing curves, circles, etc.]]
+  * [[Polygons]]
-====Angles and Vectors====
-__Complementary angles__ add up to 90' (like the two non-right angles in a right triangle)
-__Supplementary angles__ add up to 180'
-__Distance between two points__ = √(Δx² + Δy²)  (based on Pythagorean theorem)
-Length of a vector can be written as ||v|| (the "norm" of vector v)<p>
-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)
-====Polygons====
-The __sum of angles in a polygon__ = 180 (n - 2), where n = number of sides. This can also be written as: 180 + 180 (n-3). (Basically you add in another triangle when adding a side to polygon)
-====Triangles====
-Definitions:
-  * isosceles - 2 sides the same length
-  * equilateral - all 3 sides the same length
-  * similar triangles - triangles that have same shape (all 3 angles), but size can be different. The sides have same ratios
-  * congruent triangles - triangles that have the same shape and size
-  * CPCT - corresponding parts of congruent triangles (are equal)
-__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
-Area of a triangle = base*height/2
-__Heron's formula__ - area of any triangle, with sides of length a,b,c, don't know height
-  semiperimeter (sp) = perimeter/2 = (a+b+c)/2
-  area = √(sp(sp-a)(sp-b)(sp-c))
-===Congruent triangles===
-Theorems to prove triangles are congruent:
-  * AAS - angle-angle-side
-  * ASA - angle-side-angle
-  * SAS - side-angle-side
-  * SSS - side-side-side
-  * RHS - right-angle, hypotenuse, side (basically ASS, which doesn't work for all angles, but does for right-angle). Using Pythagorean theorem can be converted to SSS.
-===Right Triangles===
--45-90 triangle: hyp = side * √2
--60-90 triangle: short side = a, long side = a*√3, hyp = 2a
-__area__ of a right triangle = 1/2*h*w
-__hypotenuse__ is the side opposite the right angle, __opposite__ is the side opposite the given angle, __adjacent__ is the side next to the given angle
-SOHCAHTOA (Soak a toe-ah): Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj
-__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) = (b² + c² - a²) / (2bc)
-  cos(C) = (a² + b² - c²) / (2ab), cos(B) is same pattern
-    (side a is opposite angle A, etc)
-  rewriting it: c = √(a² + b² - 2abcos(C))
-====Circles====
-  * __chord__ has two endpoints on a circle
-  * __secant__ is a line that contains a chord, but extends beyond the circle
-  * __Intercepted arc__ is the part of the circle contained within the two lines
-  * __Central angle__ is 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__ is same as the degrees of intercepted arc
-if two chords AB, CD intersect at P, then AP * PB = CP * PD
-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*r²/2  .... comes from a = πr² for full circle, and the
-                      proportion of a circle in the arc is rad/2π
-                      so a = (rad/2π)*(πr²)
-====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 = r²
-  where point (h,k) is the center, and r is radius
-  This can be expanded to x² + y² + 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/a² + (y-k)²/b² = 1
-  and
-     c² = a² - b²
-  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
 Back to [[math]] page.

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