Question

the triangle below is equilateral. find the length of side ( x ) in simplest radical form with a rational denominator.
(image of an equilateral triangle with a perpendicular segment of length ( sqrt{12} ) to one side, forming a right triangle, and side ( x ) labeled on the equilateral triangles side)

Explanation:

Step1: Analyze the equilateral triangle

In an equilateral triangle, the altitude splits it into two 30 - 60 - 90 right triangles. The altitude (here $\sqrt{12}$) is opposite the 60° angle, and the side of the equilateral triangle ($x$) is the hypotenuse of the right triangle. In a 30 - 60 - 90 triangle, the ratio of the sides is $1:\sqrt{3}:2$, where the side opposite 30° is the shortest leg, opposite 60° is $\sqrt{3}$ times the shortest leg, and the hypotenuse is twice the shortest leg. Also, we know that $\sin(60^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{\sqrt{12}}{x}$. Since $\sin(60^{\circ})=\frac{\sqrt{3}}{2}$, we can set up the equation $\frac{\sqrt{3}}{2}=\frac{\sqrt{12}}{x}$. First, simplify $\sqrt{12}=\sqrt{4\times3} = 2\sqrt{3}$.

Step2: Solve for x

Substitute $\sqrt{12}=2\sqrt{3}$ into the equation $\frac{\sqrt{3}}{2}=\frac{2\sqrt{3}}{x}$. Cross - multiply: $\sqrt{3}\times x=2\times2\sqrt{3}$. So $\sqrt{3}x = 4\sqrt{3}$. Divide both sides by $\sqrt{3}$ (assuming $\sqrt{3}
eq0$), we get $x = 4$. Alternatively, using the 30 - 60 - 90 triangle ratio: the altitude (opposite 60°) is $\sqrt{3}$ times the shorter leg. Let the shorter leg be $a$, then $\sqrt{12}=\sqrt{3}a$. Solve for $a$: $a=\frac{\sqrt{12}}{\sqrt{3}}=\frac{2\sqrt{3}}{\sqrt{3}} = 2$. Then the hypotenuse (which is the side of the equilateral triangle $x$) is $2a=2\times2 = 4$.

Answer:

$4$