30-60-90 Triangles                    Page 4 of 5

The Shortcut

The long leg is the same as the short leg times the square root of 3.  This relationship is true of every 30-60-90 triangle.  So from now on, don't use the Pythagorean Theorem.  Use the shortcut.  If you know the short leg, just multiply it by the square root of 3 to find the long leg.

Example 1
Find c and d.

Answer:
It might help to sketch out the rest of the equilateral triangle so you can visualize the relationships between the sides.  Remember, the two legs form the letter L that makes the 90 degree angle.  We've been given the shorter leg.

We know the hypotenuse is twice as large as the short leg.  This gives us d = 18.
To find the longer leg, we multiply the short let by the square root of 3. This means c = 9 times the square root of 3. 
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Example 2
Find e and f.

Answer:
We've been given the hypotenuse of the 30-60-90 triangle.  We know that the short leg is always half the length of the hypotenuse.  Half of 26 is 13 so e = 13.

To find the long leg, we just multiply the short leg by the square root of 3. This means f = 13 times the square root of 3.

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