Question

Let ABC be a triangle having the ratios of the length of the two side sharing the common vertex A as 2:3 . Let AM be the media and Ak be the angle bisector of the triangle. Find the ratio of the areas of the triangle AKM and the triangle AKB.

           ( Giải bằng tiếng anh )

Answer 1

On the supposition that AB<AC

AK be the angle bisector of the triangle

\(\Rightarrow\)  \(\frac{KB}{KC}=\frac{AB}{AC}=\frac{2}{3}\)

\(\Rightarrow\frac{MB-MK}{MC+MK}=\frac{MC-MK}{MC+MK}=\frac{2}{3}\)

\(\Rightarrow3MC-3MK=2MC+2MK\)

\(\Rightarrow MC=5MK\)

\(\Rightarrow BK=MC-MK=5MK-MK=4MK\)

Let AH be the height of the triangle

\(\Rightarrow\frac{S_{AKM}}{S_{ABK}}=\frac{\frac{AH.KM}{2}}{\frac{BK.AH}{2}}=\frac{KM}{4KM}=\frac{1}{4}\)

If AB > AC then

\(\Rightarrow CM=5MK\)

\(\Rightarrow Bk=CM+MK=5MK+MK=6MK\)

\(\Rightarrow\frac{S_{AKM}}{S_{AKB}}=\frac{\frac{AH.MK}{2}}{\frac{AH.BK}{2}}=\frac{MK}{6MK}=\frac{1}{6}\)