in a 30-60-90 right triangle, the hypotenuse is 12 cm. What is the length of the longer of the two legs?

Question

it's a special right triangle. 

Answer 1

If you don't know Trigonometry, in a 30-60-90 triangle, the hypotenuse is always twice as long as the side opposite the 30° angle and the side opposite the 60° angle is √3 times as long as the side opposite the 30° angle.  

 

So, since the hypotenuse has length 12 cm, 

 

     side opposite 30° angle (shorter leg) = 6 cm

 

     side opposite 60° angle (longer leg) = 6√3 cm

 

 

Answer 2

There are a few "special right triangles" that appear very often on homework and on tests (especially standardized tests).  You should learn them:  They include:

         45° - 45° - 90°       with sides of      n, n, n√2

         30° - 60° - 90°       with sides of      n, n√3, 2n

          (learn later)           with sides of      3n, 4n, 5n,

          (learn later)           with sides of      5n, 12n, 13n

The numbers (3,4,5) and (5,12,13) are called "Pythagorean triples."

 

This problem mentions a 30° - 60° - 90° triangle with sides of length n, n√3, 2n.  The longest side, the hypotenuse, is 12 cm (so, n=6 cm) and the longer leg is 6√3 cm.

Answer 3

The longer of the two legs will be opposite the larger angle (60 degrees)

 

sin 60 = opposite side/hypotenuse

 

sin 60 = x/12 = √3/2 

 

x = 6√3

Answer 4

In this type of triangle, the shorter of the two legs is one half of the hypotenuse (6, in this case), and that short leg is multiplied by the square root of 3 to obtain the length of the longer leg: 6√3. 

 

Note: shorter leg is 'opposite' the 30^o angle.