# The Ungerecht Family Tree

urp:triangles

## 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:

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

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))```

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