In figure,$E$ and $F$ are mid - points of the sides $AB$ and $AC$ respectively of the $\Delta ABC$. $G$ and $H$ are mid-points of the sides $AE$ and $AF$ respectively of the $\Delta AEF$. If $GH=1.8cm,$ find $BC.$

The correct option is D: $7.2cm$

By Midpoint theorem: The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

$\because$ E and F are mid-points of sides $AB$ and $AC$ of $\Delta ABC$

$\therefore EF\parallel BC,$ and $EF=\frac{1}{2}BC\dots \dots \left(i\right)$

$\because$ $G$ and $H$ are mid-points of sides $AB$ and $AC$ of $\Delta AEF$

$GH\parallel EF,GH=\frac{1}{2}EF\dots \dots \left(ii\right)$

From $eq.\left(ii\right)$ and $\left(ii\right)$, we have,

$GH=\frac{1}{2}\times \frac{1}{2}BC=\frac{1}{4}BC$

$\Rightarrow BC=4\times GH$

$=4\times 1.8cm=7.2cm$ [$GH=1.8cm$ given]

Hence, $BC=7.2cm$