In the figure above, triangles $ABC$ and $CDE$ are equilateral and line segment $\overline{AE}$ has length $35$. What is the sum of the perimeters of the two triangles?

เฉลยละเอียด

[STEP]Express the length of $\overline{AE}$ in terms of $\overline{AC}$ and $\overline{CE}$[/STEP]

Since they are equilaterals so each of their sides are equal.

$$\overline{AE} = \overline{AC} +\overline{CE}$$

[STEP]Calculate the sum of perimeters[/STEP]

\begin{eqnarray*}
\left(\overline{AB}+\overline{BC}+\overline{AC}\right)+\left(\overline{CD}+\overline{DE}+\overline{CE}\right) & = & 3\left(\overline{AC}\right)+3\left(\overline{CE}\right)\\
& = & 3\left(\overline{AC}+\overline{CE}\right)\\
& = & 3\left(\overline{AE}\right)\\
& = & 3\times35\\
& = & 105
\end{eqnarray*}

[ANS]$105$[/ANS]