Respuesta :
An equilateral triangle is a triangle were all the sides have the same measurement, and all the angles are the same(60º).
The height of an equilateral triangle divides the triangle into two equal right triangles. The height represents the oposite side of the angle of 60º, and the hypotenuse has the length of the side of the equilateral triangle, if we find the hypotenuse we have our answer.
Using trigonometric relations on the right triangle, we can find the value for the hypotenuse. The ratio between the opposite side to an angle and the hypotenuse is equal to the sine of this angle. If we call the hypotenuse as h, we have the following relation
[tex]\sin (60^o)=\frac{9\sqrt[]{3}}{h}[/tex]The sine of 60º is a known value
[tex]\sin (60^o)=\frac{\sqrt[]{3}}{2}[/tex]Then, combining both expressions, we have
[tex]\frac{9\sqrt[]{3}}{h}=\frac{\sqrt[]{3}}{2}[/tex]Solving for h
[tex]\begin{gathered} \frac{9\sqrt[]{3}}{h}=\frac{\sqrt[]{3}}{2} \\ \frac{9}{h}=\frac{1}{2} \\ \frac{h}{9}=2 \\ h=18 \end{gathered}[/tex]The length of the side of an equilateral triangle if the height is 9√3 is equal to 18.
