a.
Trong \(\Delta ADC\) do \(CD||MN\) hay \(CD||MP\), áp dụng định lý Talet:
\(\dfrac{AM}{MD}=\dfrac{AP}{PC}\) (1)
Tương tự, trong \(\Delta ABC\) do \(AB||PN\) nên: \(\dfrac{AP}{PC}=\dfrac{BN}{NC}\) (2)
(1);(2) \(\Rightarrow\dfrac{AM}{MD}=\dfrac{BN}{NC}\)
b.
Ta có: \(MD=2MA\Rightarrow AD-MA=2MA\Rightarrow AD=3MA\Rightarrow\dfrac{MA}{AD}=\dfrac{1}{3}\)
Áp dụng định lý Talet trong tam giác ACD:
\(\dfrac{MA}{AD}=\dfrac{MP}{CD}=\dfrac{1}{3}\Rightarrow MP=\dfrac{CD}{3}=\dfrac{6}{3}=2\left(cm\right)\)
Lại có: \(\dfrac{BN}{NC}=\dfrac{AM}{MD}=\dfrac{1}{2}\Leftrightarrow NC=2BN\Rightarrow NC=2\left(BC-NC\right)\)
\(\Rightarrow3NC=2BC\Rightarrow\dfrac{NC}{BC}=\dfrac{2}{3}\)
Áp dụng định lý Talet cho tam giác ABC:
\(\dfrac{PN}{AB}=\dfrac{BC}{BC}=\dfrac{2}{3}\Rightarrow PN=\dfrac{2}{3}AB=\dfrac{8}{3}\left(cm\right)\)
\(\Rightarrow MN=MP+PN=\dfrac{14}{3}\left(cm\right)\)
