A surveyor is examining a triangular framework \(\triangle ABC\). A line \(DE\) is drawn inside the triangle such that \(D\) lies on \(AB\), \(E\) lies on \(AC\) and \(DE \parallel BC\).
The following measurements are recorded : \(AD\ =\ 16 cm\), \(DB\ =\ 24 cm \) and \(AE\ =\ 12 cm \), \(EC\ =\ 18 cm\).
Later, another point \(F\) is marked on \(AB\) such that \(AF\ =\ 12 cm \), \(FB\ =\ 18 cm \). A point \(G\) marked on \(AC\) such that \(AG\ =\ 12 cm \), \(GC\ =\ 18 cm \).
Answer the following questions:
1. Find the ratio of \(\frac{AD}{DB}\) and \(\frac{AE}{EC}\). Verify whether the ratios are equal or not.
Therefore, the ratios are
2. For points \(F\) and \(G\) determine whether \(FG \parallel BC \). Also determine the theorem used to find whether they are parallel or not.
3. Find the ratio of \(DE : BC \) if \( \triangle ADE \sim \triangle ABC \)
4. Find the ratio of the perimeter of \(\triangle ADE\) and \(\triangle ABC\) if \( \triangle ADE \sim \triangle ABC \)