Express the area of an equilateral triangle


Express the area of an equilateral triangle as a function of the length of a side.

Solution:

Let “x” represent the side length of the equilateral triangle. If we let “h” represent the height of the  triangle, we can use Pythagorean theorem to figure it out.

Thus,

h^{2}+(\frac{1}{2}x)^{2}=x^{2}

Expanding this, we get

h^{2}=x^{2}-\frac{1}{4}x^2=\frac{3}{4}x^{2}

and solving for h, we have

h=\frac{\sqrt{3}}{2}x

The area of a triangle is equal to A=\frac{1}{2}(base)(height). If we write area as a function of length, then we have

A(x)=\frac{1}{2}(x)(\frac{\sqrt{3}}{2}x)=\frac{\sqrt{3}}{4}x^{2} and the domain is x>0.

 

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