Question

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?

Answer 1

Since it's a 30-60-90 triangle, the hypotenuse should be `6sqrt(3)` and the short leg is `3sqrt(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(60)="short"/"hyp"=1/2 rArr "short"=1, "hyp"=2`

`sin(60)="long"/"hyp"=sqrt(3)/2 rArr "long"=sqrt(3), "hyp"=2`

`tan(60)="long"/"short"=sqrt(3) rArr "long"=sqrt(3), "short"=1`

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

`"short"=1x=x`

`"hyp"=2x`

`"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 rArr x=9/sqrt(3)=3*sqrt(3)^2/sqrt(3)`

`color(red)(x=3sqrt(3))`

`color(blue)("short"=x=3sqrt(3))`

`color(green)("hyp"=2x=6sqrt(3))`

Answer 2

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`