In \(\triangle ABC\), F and E are points on the sides AB and AC respectively such that \(FE\parallel BC\) and FE divides the triangle into two parts of equal area. AD is perpendicular to BC and cuts FE at G. What is GD : AG?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((\sqrt{2}-1):1\)
Since \(FE\parallel BC\), \(\triangle AFE\) and \(\triangle ABC\) are similar. The ratio of the areas of similar triangles equals the square of the ratio of corresponding lengths, and their heights AG and AD are corresponding lengths.
FE halves the area, so area of \(\triangle AFE\) : area of \(\triangle ABC=1:2\)
\(\displaystyle\left(\frac{AG}{AD}\right)^2=\frac{1}{2}\), so \(\displaystyle\frac{AG}{AD}=\frac{1}{\sqrt{2}}\)
Take AG \(=1\) and AD \(=\sqrt{2}\); then GD \(=\sqrt{2}-1\)
GD : AG \(=(\sqrt{2}-1):1\)
