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urp:geometry [2021-10-13]
nerf_herder [Triangles]
urp:geometry [2021-11-17] (current)
nerf_herder
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 =====Geometry Reference Page===== =====Geometry Reference Page=====
  
-Contents: +  ​* [[Angles|Angles and Vectors]] 
-  ​* [[#Angles and Vectors]] +  * [[Triangles]] 
-  * [[#​Polygons]] +  * [[Circles|CirclesChords, etc]] 
-  * [[#Triangles|Right triangles]] +  * [[Polygons]] 
-  * [[#Circles|Circles ​and Chords]] +  
-  * [[#​Graphing|Graphing curvescirclesetc.]]+=== misc === 
 +Truth tables: 
 +first columns : True/False values of the variablessuch as pq, or p' or ~p for inverse valuesNext columns are logic combinations of the variables
  
-====Angles and Vectors==== +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) +F | T | F 
- +F | T | F | T | T 
-__Supplementary angles__ add up to 180' +| F | | F | F | T |
- +
-__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) +
- +
-__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. +
- +
-__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 = √(sp(sp-a)(sp-b)(sp-c)) +
- +
-Congruent triangles +
- +
-===Right Triangles=== +
- +
-45-45-90 triangle: hyp = side * √2 +
-30-60-90 triangle: short side = a, long side = a*√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) = (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)) +
-</​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. ​  +
-      2pi - |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) +
-</​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 = r² +
-  where point (h,k) is the center, and r is radius +
-  This can be expanded to x² + y² + Dx + Ey + = 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 falsethen an implication cannot be ruled out, regardless ​of qso it is left as true.
-  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 
-</​nowiki>​ 
  
 === See Also === === 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   * 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. Back to [[math]] page.
  
urp/geometry.1634090787.txt.gz · Last modified: 2021-10-13 by nerf_herder