SOLUTION: how can i solve a Pythagorean theorem of an equilateral triangle if the only given is the altitude of the triangle???answer please,,,,

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Question 208677: how can i solve a Pythagorean theorem of an equilateral triangle if the only given is the altitude of the triangle???answer please,,,,
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
"how can i solve a Pythagorean theorem of an equilateral triangle...." The Pythagorean Theorem applies only to right triangles so this does not make sense.

But I will guess the problem is to find the sides of the equilateral triangle when all you are given is the altitude. Solving this may involve the Pythagorean Theorem, not on the equilateral triangle, but on one of the right triangles formed by the altitude.

Since the three sides of an equilateral triangle are equal the three interior angles of the triangle are also equal. And since the 3 interior angles of every triangle add up to 180 degrees, these three equal angles of equilateral triangles must be 60 degrees each. (3*60 = 180). Let's look at a diagram of an equilateral triangle ABC with an altitude, AD, draw from A:

Since altitudes by definition are perpendicular to the side, the angles at D are right angles:

Now we can tell that the two right triangles, ADB and ADC, are 30-60-90 right triangles. At this point, if you understand how to use the ratios of 30-60-90 right triangles, you can use one side, the altitude, to find the other two sides. But since you referred to the Pythagorean Theorem I'll explain the solution using the theorem.

Since AC is congruent to AB and AD is congruent to itself, these two right triangles are congruent 30-60-90 right triangles. Therefore BD is congruent to DC. And, since BD = DC and since BD + DC = BC, 2*DC = AC.

The Pythagorean theorem for triangle ADC would be:
%28AD%29%5E2+%2B+%28DC%29%5E2+=+%28AC%29%5E2
Since AC = 2*DC I will substitute for AC:
%28AD%29%5E2+%2B+%28DC%29%5E2+=+%282%2ADC%29%5E2
Simplifying we get:
%28AD%29%5E2+%2B+%28DC%29%5E2+=+4%28DC%29%5E2
Subtracting (DC)^2 from both sides:
%28AD%29%5E2+=+3%28DC%29%5E2
Dividing both sides by 3:
%28%28AD%29%5E2%29%2F3+=+%28DC%29%5E2
Find the square root of both sides:
sqrt%28%28%28AD%29%5E2%29%2F3%29+=+sqrt%28%28DC%29%5E2%29
%28AD%29%2F%28sqrt%283%29%29+=+DC
Rationalizing the denominator on the left:
%28%28AD%29%2F%28sqrt%283%29%29%29%2A%28%28sqrt%283%29%29%2F%28sqrt%283%29%29%29+=+DC
%28%28AD%29%2Asqrt%283%29%29%2F3+=+DC
%28%28sqrt%283%29%29%2F3%29%2A+AD+=+DC
Since AD is the altitude and DC is one of the sides of the equilateral triangle, this equation tells that that if you mulitply the length of the altitude by sqrt%283%29%2F3 you will get the length of the side of the equilateral triangle.