SOLUTION: what is the length of an altitude of an equilateral triangle with side lenghts 8 radical 3

Question 642302: what is the length of an altitude of an equilateral triangle with side lenghts 8 radical 3

Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
An equilateral triangle has three equal angles of 60 degrees. The height is the lenght of the line drawn from one of the vertices to be perpendicular to the opposite side. This line (height) forms a 30, 60, 90 degree triangle. The sine of the 60 degree angle is the ratio of the height to the given side of the equilateral triangle. Using your problem statement we have
(1) sin(60) = h/(8*sqrt(3)) or
(2) h = 8*(sqrt(3))*sin(60)
Now use the value for
(3) sin(60) = (sqrt(3))/2 in (2) to get
(4) h = 8*((sqrt(3))^2)/2 or
(5) h = 8*3/2
(6) h = 12
You can also use a calculator the evaluate (2) and get the same answer. I just don't use a calculator for this type of problem. I caution you not to depend on their use, students make more mistakes with them, but more importantly you may not know if your answer is correct or not.
In fact, how do we know if 12 is correct? Let's check it.
Since the height forms a right triangle it should obey Pythagorean's theorem.
Is (c^2 = a^2 + b^2)?
Is ((8*sqrt(3))^2 = 12^2 + ((8*sqrt(3)/2)^2)?
Is (64*3 = 144 + 64*3/4)?
Is (192 = 144 + 48)?
Is (192 = 192)? Yes
Note that b above is one half c, the lenght of the side of the equilateral triangle.
Answer: The lenght of the altitude is 12 units.
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