SOLUTION: What is the area of an equilateral triangle whose altitude is 9?

Question 119312: What is the area of an equilateral triangle whose altitude is 9?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
If you construct the altitude of an equilateral triangle, you create a 30-60-90 degree right triangle.

If you know that the proportions of the sides of a 30-60-90 right triangle are ::, fine. Otherwise, read the following discussion.

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Assume you have an equilateral triangle with sides that measure 1. Construct the altitude. The altitude bisects one of the sides, so the short leg of the resulting 30-60-90 triangle is .

. So the sides are in proportion ::.
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In the given equilateral triangle, the hypotenuse of the 30-60-90 right triangle is equal to the base of the equilateral triangle because the equilateral triangle's sides are all equal. So to find the measure of the base of the equilateral triangle we need the proportion:

Cross-multiply:

Multiply by

Rationalize the denominator:

The area of the triangle is given by where b is the base and h is the height or altitude. Substituting our given and calculated values:

square units.

Hope that helps,
John
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