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Sides of a 30-60-90 Triangle

Date: 03/13/98 at 11:03:56
From: Susan Totten
Subject: Math Question

If there is a 30:60:90 triangle, and the short side is X, then why 
does the hypotenuse equal two X, and the long side equal X times the 
square root of 3? Derive this in two ways.

Date: 03/16/98 at 15:44:48
From: Doctor Sorelle
Subject: Re: Math Question

Dear Susan,

I'm going to show you one way of deriving this. Maybe from this you'll 
get an idea of how to do it in another way.

If you put two 30-60-90 triangles back to back, you get an equilateral 
triangle with a line down the middle perpendicular (at a right angle) 
to the base of the triangle.

        / | \
       /  |  \
      /   |   \

If you say that each of the sides of the triangle is 2 units (you 
could also call this 2X), then the base will also be two units. The 
line down the middle of the triangle perpendicular to the base divides 
the base into two equal parts, each with a length of 1 unit. So now 
that we know that half of the base is 1 unit and any other side is 2 
units let's separate the triangles again.

        / |
    2  /  |
      /   | ? units
     /    |

We now know 2 of the sides' lengths, so we can use the Pythagorean 
theorem (a^2+b^2 = c^2) to find the 3rd side.

Think you can do that on your own?

To do the other proof you might consider using the area of a triangle, 
Pythagorean's theorem, equilateral triangles, or other things.

Good luck - and if you need more help, please write back!

-Doctor Sorelle,  The Math Forum
Check out our web site!   
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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