[tex]\displaystyle \\
MA = MB ~~~si~~~ QD=QC\\
\Longrightarrow~~ MQ ~\text{ este linie mijlocie in trapezul }~ABCD \\
NB=ND~~~si~~~PA=PC \\
\Longrightarrow~~ N = BD \cap MQ ~~~si~~~ P = AC \cap MQ \\
\Longrightarrow~~ M, N,P,Q ~\text{ sunt coliniare} \\
MN~\text{ este linie mijlocie in }\Delta ABD \\
PQ~\text{ este linie mijlocie in }\Delta ACD \\
MP~\text{ este linie mijlocie in }\Delta ABC \\
NQ~\text{ este linie mijlocie in }\Delta BDC [/tex]
[tex]\displaystyle \\
\text{Rezolvare: } \\ \\
a) AD=4~cm ~~~si~~~BC=10~cm\\ \\
MN= \frac{AD}{2} =\frac{4}{2} = \boxed{2~cm }\\ \\
MP= \frac{BC}{2} =\frac{10}{2} = \boxed{5~cm } \\ \\
MQ= \frac{AD+BC}{2} =\frac{4+10}{2} =\frac{14}{2} = \boxed{7~cm }\\ \\
NP= MP - MN = 5 - 2=\boxed{3~cm }[/tex]
[tex]\displaystyle \\
b) AD=6~cm ~~~si~~~BC=9~cm\\ \\
NQ= \frac{BC}{2} = \frac{9}{2} = \boxed{4,5~cm}\\
MQ= \frac{AD+BC}{2} = \frac{6+9}{2} = \frac{15}{2} =\boxed{7,5~cm}\\
PQ= \frac{AD}{2} = \frac{6}{2} = \boxed{3~ cm}\\
NP= NQ-PQ = 4,5 - 3 = \boxed{1,5~cm}
[/tex]
