Properties of 30°-60°-90° Triangles
Apply 30-60-90 triangle properties to find side lengths: the short leg is half the hypotenuse and the long leg equals the short leg times sqrt(3), enabling rapid trig calculations.
Key Concepts
In a $30° 60° 90°$ triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is the shorter leg's length times $\sqrt{3}$. The side ratios follow a $1 \sqrt{3} 2$ pattern.
If the shorter leg is $5$, the hypotenuse is $2 \times 5 = 10$ and the longer leg is $5\sqrt{3}$. If the hypotenuse is $14$, the shorter leg is $\frac{14}{2} = 7$ and the longer leg is $7\sqrt{3}$. If the longer leg is $6\sqrt{3}$, the shorter leg is $6$ and the hypotenuse is $12$.
This triangle is an equilateral triangle neatly cut in half. The shortest leg (opposite the $30°$ angle) is your key. The hypotenuse is always double the short leg, and the longer leg (opposite the $60°$ angle) is the short leg multiplied by $\sqrt{3}$. This simple relationship makes finding side lengths a total breeze!
Common Questions
What are the side length ratios of a 30-60-90 triangle?
In a 30-60-90 triangle, if the short leg (opposite the 30-degree angle) has length n, then the hypotenuse is 2n and the long leg (opposite the 60-degree angle) is n*sqrt(3). This fixed ratio applies to any 30-60-90 triangle regardless of size.
How do you find the missing sides of a 30-60-90 triangle given the hypotenuse?
Divide the hypotenuse by 2 to get the short leg. Multiply the short leg by sqrt(3) to get the long leg. For a hypotenuse of 10: short leg = 5, long leg = 5*sqrt(3) which is approximately 8.66.
Where do 30-60-90 triangles appear in Grade 10 Saxon Algebra 2?
These triangles appear when evaluating exact trig values at 30 and 60 degrees, in problems involving equilateral triangles cut in half, in altitude calculations, and whenever a triangle with these angles appears in geometry or trigonometry sections.