The short side of the 30-60-90 degree triangle is opposite the 30 degree angle and the long side is opposite the 60 degree angle. The hypotenuse is opposite the 90 degree angle.
The lengths of the sides are based on the short side. If we call the short side a, the long side b, and the hypotenuse c, then a = a, b = a
c = 2a
Suppose a = 4 units. Then the hypotenuse is 2(4) or 8 units and the long leg is 4
In a 45-45-90 degree triangle the legs that intersect to form the 90 degree angle are congruent. Therefore, a = b. The length of the hypotenuse, the leg opposite the 90 degree angle is a
Suppose the length of the legs is 5. Then the hypotenuse is 5
Sometimes we know the length of the hypotenuse rather than the length of the side or sides.
When this happens in a 45-45-90 degree triangle, we have to use the formulas c = a
Suppose the length of the hypotenuse is 9. To find a we substitute 9 for c and solve the equation.
c = a
9 = a
Divide both sides of the equation by
Now we have to rationalize the denominator because we cannot have a square root for a denominator. We do this by multiplying the fraction
by an equivalent of 1 that will eliminate the square root.
is such a fraction.
The length of the sides is
To prove this use the pythagorean theorem.
81(2)/4 + 81(2)/4 = 81
81/2 + 81/2 = 81
162/2 = 81
To find the short side of a 30-60-90 degree triangle when given the length of the long side, follow the same steps given above, but use b = a
instead of c = a