Difference between revisions of "30-60-90 triangle"

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A '''30-60-90 triangle is a right triangle with a 30 degree angle, a 60 degree angle,a 90 degree angle. It is special because it side lengths are always in the same ratio. The length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is <math>sqrt{3}</math> the length of the shorter leg.
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A '''30-60-90''' triangle is a right triangle with a 30 degree angle, a 60 degree angle,a 90 degree angle. It is special because it side lengths are always in the same ratio. The length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is <math>\sqrt{3}</math> the length of the shorter leg.
  
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Simply put, the ratio in order of the sides opposite to the angles for any value of <math>y</math> is <math>y</math> : <math>y</math><math>\sqrt{3}</math> : <math>2y</math>. It is a simple "pneumonic" ratio that will help you remember the ratio and each part corresponds to the part of the 30-60-90 triangle. (For example: the <math>y\sqrt{3}</math> corresponds to the <math>60°</math> angle)
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[[Image:30-60-902.PNG|center]]
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==Proofs==
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Consider a right triangle with angles 30, 60, and 90. Let:
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* a: side opposite the 30 angle
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* b: side opposite the 60 angle
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* c: hypotenuse
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===Using trigonometric identities===
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==== 1. Hypotenuse <math>AC</math> ====
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Here are two pieces of evidence:
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sin(30)=oppositehypotenuse=ABAC
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sin(30)=12
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Setting them equal to each other, we have:
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<math>\boxed{\frac{{AB}}{{AC}} = \frac{1}{2}  \text{ or } \bf{2AB=AC}}</math>
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==== 2. Side <math> AB </math> ====
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cos(30)=adjacenthypotenuse=BCAC
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Since cos(30)=32:
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<math>\frac{\sqrt{3}}{2} = \frac{BC}{AC} \implies BC = AC \cdot \frac{\sqrt{3}}{2} \implies \boxed{\bf{BC = AB \cdot \sqrt{3}}}</math>
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In conclusion, the side lengths are:
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\begin{array}{l}
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\text{Opposite } 30^\circ: AB \
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\text{Opposite } 60^\circ: AB\sqrt{3} \
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\text{Hypotenuse }    : 2AB
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\end{array}
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Therefore, the side ratios are <math>\boxed{1:\sqrt{3}:2}</math>

Latest revision as of 16:37, 30 January 2025

A 30-60-90 triangle is a right triangle with a 30 degree angle, a 60 degree angle,a 90 degree angle. It is special because it side lengths are always in the same ratio. The length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is $\sqrt{3}$ the length of the shorter leg.

Simply put, the ratio in order of the sides opposite to the angles for any value of $y$ is $y$ : $y$$\sqrt{3}$ : $2y$. It is a simple "pneumonic" ratio that will help you remember the ratio and each part corresponds to the part of the 30-60-90 triangle. (For example: the $y\sqrt{3}$ corresponds to the $60°$ angle)

30-60-902.PNG


Proofs

Consider a right triangle with angles 30, 60, and 90. Let:

  • a: side opposite the 30 angle
  • b: side opposite the 60 angle
  • c: hypotenuse

Using trigonometric identities

1. Hypotenuse $AC$

Here are two pieces of evidence:

sin(30)=oppositehypotenuse=ABAC

sin(30)=12


Setting them equal to each other, we have:

$\boxed{\frac{{AB}}{{AC}} = \frac{1}{2}  \text{ or } \bf{2AB=AC}}$


2. Side $AB$

cos(30)=adjacenthypotenuse=BCAC

Since cos(30)=32:

$\frac{\sqrt{3}}{2} = \frac{BC}{AC} \implies BC = AC \cdot \frac{\sqrt{3}}{2} \implies \boxed{\bf{BC = AB \cdot \sqrt{3}}}$


In conclusion, the side lengths are:


Opposite 30:ABOpposite 60:AB3Hypotenuse :2AB

Therefore, the side ratios are $\boxed{1:\sqrt{3}:2}$