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. | 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. | ||
| − | 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 | + | 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>x\sqrt{3}</math> corresponds to the <math>60°</math> angle) |
[[Image:30-60-902.PNG|center]] | [[Image:30-60-902.PNG|center]] | ||
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==== 1. Hypotenuse <math>AC</math> ==== | ==== 1. Hypotenuse <math>AC</math> ==== | ||
| + | |||
| + | Here are two pieces of evidence: | ||
\(\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC}\) | \(\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC}\) | ||
| − | + | \(\sin(30^\circ) = \frac{1}{2}\) | |
| − | |||
| − | \(\sin(30^\circ) = \frac{1}{2 | ||
| + | Setting them equal to each other, we have: | ||
<math>\boxed{\frac{{AB}}{{AC}} = \frac{1}{2} \text{ or } \bf{2AB=AC}}</math> | <math>\boxed{\frac{{AB}}{{AC}} = \frac{1}{2} \text{ or } \bf{2AB=AC}}</math> | ||
Revision as of 14:58, 4 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 
the length of the shorter leg.
Simply put, the ratio in order of the sides opposite to the angles for any value of 
is : : 
. 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 
corresponds to the 
angle)
Proofs
Consider a right triangle with angles \(30^\circ\), \(60^\circ\), and \(90^\circ\). Let:
- \(a\): side opposite the \(30^\circ\) angle
- \(b\): side opposite the \(60^\circ\) angle
- \(c\): hypotenuse
Using trigonometric identities
1. Hypotenuse 
Here are two pieces of evidence:
\(\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC}\)
\(\sin(30^\circ) = \frac{1}{2}\)
Setting them equal to each other, we have:
2. Side 
\(\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC}\)
Since \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):
In conclusion, the side lengths are:
\begin{array}{l} \text{Opposite } 30^\circ: AB \\ \text{Opposite } 60^\circ: AB\sqrt{3} \\ \text{Hypotenuse } : 2AB \end{array}
Therefore, the side ratios are 
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