Question

Find the length of side xx in simplest radical form with a rational denominator.

Answer 1

well, the triangle is an equilateral, so all sides are equal, we know that the segment "x" stems out of the vertex and drops perpendicular to the other side, now, the vertex "x" stems from has two equal sides.

what the hell all that means? well it means that "x" is an altitude and is also an angle bisector and likewise is also cutting the opposite side into two equal halves, meaning that one side is really 8 + 8 = 16, so then

`\stackrel{altitude}{height}\textit{ of an equilateral triangle}\\\\ h=\cfrac{s√(3)}{2} ~~ \begin{cases} s=\stackrel{length~of}{a~side}\\[-0.5em] \hrulefill\\ s=16 \end{cases}\implies h=\cfrac{16√(3)}{2}\implies h=8√(3)`