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?

2 Answers
Ben
Mar 27, 2018

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))#

AltairSafir
Mar 27, 2018

Use trigonometric function

Explanation:

#b=9#
#alpha=30°#
#beta=60°#
#gamma=90°#
my pic
#a=?#
#c=?#
#tan(30°)=a/b#
#a=tan(30°)b=3*√3#
#cos(30°)=b/c#
#c=b/cos(30°)=6*√3#

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