An equilateral triangle is one in which all three sides (and all three angles) are equal. Each angle of an equilateral triangle is 60° (because all three angles must sum to 180°). A close relative of the equilateral triangle is the 30–60–90 triangle. Notice that two 30–60–90 triangles, when put together, form an equilateral triangle:
The lengths of the legs of every 30–60–90 triangle have the following ratio, which you must memorize:
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If the short leg of a 30–60–90 triangle has a length of 6, what are the lengths of the long leg and the hypotenuse?
The short leg, which is opposite the 30° angle, is 6. Use the ratio
to determine that the multiplier x is 6. You then find that the sides of the triangle have lengths
. The long leg measures
and the hypotenuse measures 12. Try another problem:
If an equilateral triangle has a side of length 10, what is its height?
Looking at the equilateral triangle above, you can see that the side of an equilateral triangle is the same as the hypotenuse of a 30–60–90 triangle. Additionally, the height of an equilateral triangle is the same as the long leg of a 30–60–90 triangle.
Because you are told that the hypotenuse is 10, use the ratio
to get 2x = 10 and determine that the multiplier x is 5. You then find that the sides of the 30–60–90 triangle have lengths
. Thus, the long leg has a length of
, which is the height of the equilateral triangle.
If you get tangled up on a 30–60–90 triangle, try to find the length of the short leg. The other legs will then be easier to figure out.
Quadrilateral ABCD is composed of four 30–60–90 triangles. If
, what is the perimeter of ABCD?
Each side of the equilateral triangle shown is 2. What is the height h of the triangle?
