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urp:triangles
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urp:triangles [2021-10-15] nerf_herder |
urp:triangles [2021-11-08] (current) nerf_herder |
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| - | ====Triangles==== | + | =====Triangles===== |
| Definitions: | Definitions: | ||
| - | * isosceles - 2 sides the same length | + | * isosceles - (at least) 2 sides the same length, modern definition includes equilateral |
| * equilateral - all 3 sides the same length | * 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 | * 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 | * congruent triangles - triangles that have the same shape and size | ||
| * CPCT - corresponding parts of congruent triangles (are equal) | * CPCT - corresponding parts of congruent triangles (are equal) | ||
| - | __Centroid of a triangle__ - center (where the lines bisecting each angle will meet) | + | 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 | * 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 (basically ASS, which doesn't work for acute angles, but does for right-angle). Using Pythagorean theorem can be converted to SSS. | + | * 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=== | + | ====Right Triangles==== |
| - | 45-45-90 triangle: hyp = side * √2 | + | * 45-45-90 triangle: hyp = side * √2 |
| - | 30-60-90 triangle: short side = a, long side = a*√3, hyp = 2a | + | * 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 | + | **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.1634315712.txt.gz · Last modified: 2021-10-15 by nerf_herder
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