The Ungerecht Family Tree

Contents:

• Angles and Vectors
• Polygons
• Right triangles
• Circles and Chords
• Graphing curves, circles, etc.

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)

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)

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))

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

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))

• 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.  
    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)

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

• 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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