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urp:triangles [2021-10-28]
nerf_herder
urp:triangles [2021-11-08] (current)
nerf_herder
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-====Triangles====+=====Triangles=====
  
 Definitions:​ Definitions:​
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   * CPCT - corresponding parts of congruent triangles (are equal)   * CPCT - corresponding parts of congruent triangles (are equal)
  
-__Isosceles triangle__ ​- base angles are the same, __Equilateral__ ​- all angles are the same (60°), aka equiangular+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)+**Centroid ​of a triangle** ​- center (where the lines bisecting each angle will meet)
   * average the x corner values, and average of y corner values   * 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 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
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 Area of a triangle = base*height/​2 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:+**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   - Find semiperimeter (sp) = perimeter/2 = (a+b+c)/2
   - area = √(sp(sp-a)(sp-b)(sp-c))   - area = √(sp(sp-a)(sp-b)(sp-c))
  
-===Congruent triangles===+====Congruent triangles====
  
 Theorems to prove triangles are congruent: Theorems to prove triangles are congruent:
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   * SAS - side-angle-side   * SAS - side-angle-side
   * SSS - side-side-side   * SSS - side-side-side
-  * RHS - right-angle,​ hypotenuse, side, also called HL (Hypotenuse ​& leg).  +  * 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.
-RHS is basically ASS, which doesn'​t work for acute angles, but does for right-angle. Using Pythagorean theorem it can be converted to SSS.+
  
-===Right Triangles===+====Right Triangles====
  
   * 45-45-90 triangle: hyp = side * √2   * 45-45-90 triangle: hyp = side * √2
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   * 3-4-5 triangle: if a triangle has sides 3n and 4n, then the hypotenuse will be 5n   * 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+**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+**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+**SOHCAHTOA** ​(pronounced "Soak a toe-ah"​):​ Sin=Opp/​Hyp,​ Cos=Adj/​Hyp,​ Tan=Opp/Adj
  
-__law of sines:__+**law of sines:**
   sin(A)/a = sin(B)/b = sin(C)/​c ​ (sometimes a/sin(A) = ...)   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+**law of cosines** ​- to find an angle when all the sides are known
   cos(A) = (b² + c² - a²) / (2bc)   cos(A) = (b² + c² - a²) / (2bc)
   cos(C) = (a² + b² - c²) / (2ab), cos(B) is same pattern   cos(C) = (a² + b² - c²) / (2ab), cos(B) is same pattern
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   rewriting it: c = √(a² + b² - 2abcos(C))   rewriting it: c = √(a² + b² - 2abcos(C))
  
 +------
 Back to [[geometry]] or [[math]] page. Back to [[geometry]] or [[math]] page.
  
urp/triangles.1635442984.txt.gz · Last modified: 2021-10-28 by nerf_herder