# In 30-60-90 triangle, where the length of the long leg is 9, what is the length of the hypotenuse and the short leg?

Mar 27, 2018

Since it's a 30-60-90 triangle, the hypotenuse should be $6 \sqrt{3}$ and the short leg is $3 \sqrt{3}$

#### Explanation:

In a 30-60-90 triangle, the sides can be described as such:

Short side: $1$
Hypotenuse: $2$
Long Side: $\sqrt{3}$

These can be considered ratios. If you look at it in terms of sine and cosine, this becomes a bit clearer, since sine and cosine gives you the ratio of the sides:

$\cos \left(60\right) = \text{short"/"hyp"=1/2 rArr "short"=1, "hyp} = 2$

$\sin \left(60\right) = \text{long"/"hyp"=sqrt(3)/2 rArr "long"=sqrt(3), "hyp} = 2$

$\tan \left(60\right) = \text{long"/"short"=sqrt(3) rArr "long"=sqrt(3), "short} = 1$

since we know the ratios, we can multiply them by a constant, $x$

$\text{short} = 1 x = x$

$\text{hyp} = 2 x$

$\text{long} = \sqrt{3} x = 9$

Now that we have an equation which describes the length of the long leg in terms of the side ratios, we can solve for $x$, and quickly solve for the short side and hypotenuse:

$\sqrt{3} x = 9 \Rightarrow x = \frac{9}{\sqrt{3}} = 3 \cdot {\sqrt{3}}^{2} / \sqrt{3}$

$\textcolor{red}{x = 3 \sqrt{3}}$

$\textcolor{b l u e}{\text{short} = x = 3 \sqrt{3}}$

$\textcolor{g r e e n}{\text{hyp} = 2 x = 6 \sqrt{3}}$

Mar 27, 2018

Use trigonometric function

#### Explanation:

$b = 9$
alpha=30°
beta=60°
gamma=90°

a=?
c=?
tan(30°)=a/b
a=tan(30°)b=3*√3
cos(30°)=b/c
c=b/cos(30°)=6*√3