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  • 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:

  1. Find semiperimeter (sp) = perimeter/2 = (a+b+c)/2
  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.

urp/triangles.txt · Last modified: 2021-11-08 by nerf_herder