Question

In triunghiul ABC,MN||BC,M€(AB și N € (AC.Daca MB=2radical 3 cm,CN=4 radical 6 cm,AM=3radical3,aflați lungimea segmentelor [AB],[NA],[AC].

Answer 1

AB=AM+MB=3√3+2√3=5√3
AM/AB=AN/AC
3√3/5√3=AN/(AN+NC)
3/5=AN/(AN+4√6)
5AN-3AN=12√6
AN=6√6
AC=AN+NC=6√6+4√6=10√6

Answer 2

[tex]\it MN||BC \stackrel{T. Thales}{\Longrightarrow} \dfrac{AM}{MB} =\dfrac{AN}{NC} \Longrightarrow \dfrac{3\sqrt3}{2\sqrt3}=\dfrac{AN}{4\sqrt6}\Longrightarrow\dfrac{3}{2}=\dfrac{AN}{4\sqrt6}\Longrightarrow \\\;\\ \\\;\\ \Longrightarrow AN = \dfrac{3\cdot4\sqrt6}{2} =3\cdot2\sqrt6=6\sqrt6[/tex]