=====Triangles=====

Definitions:
  * isosceles - (at least) 2 sides the same length, modern definition includes equilateral
  * equilateral - all 3 sides the same length
  * acute triangle - all angles are < 90 degrees
  * obtuse triangle - one angle is > 90 degrees
  * 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)

Isosceles triangle - base angles are the same, Equilateral - all angles are the same (60°), aka equiangular

**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:
  - Find 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, also called HL (hypotenuse & leg). 
     * RHS is basically ASS/SSA, which doesn't work for acute angles, but does for right-angle. Using Pythagorean theorem it can be converted to SSS.

====Right Triangles====

  * 45-45-90 triangle: hyp = side * √2
  * 30-60-90 triangle: short side = a, long side = a*√3, hyp = 2a
  * 3-4-5 triangle: if a triangle has sides 3n and 4n, then the hypotenuse will be 5n

**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** (pronounced "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))

------
Back to [[geometry]] or [[math]] page.