Question

Stabiliti daca triunghiurile ABC si MNP sunt asemenea, in fiecare dintre situatiile urmatoare:
a) AB= 4MN supra 5, AC=0,8•MP şi BC=80%NP
b) masura unghiului c = masura unghiului P, AB supra 3 = MN supra 4 şi BC supra 21 = NP supra 28
c) AB=6 cm, BC=9 cm, AC=12 cm, MN=12 cm, NP=9 cm, MP=18 cm
d) AB=1,2 dm, BC=1,6 dm, AC=1,8 dm, MN=0,6 dm, NP=0,8 dm, MP=0,9 dm.
e) măsura unghiului B =72 grade, măsura unghiului C = 56 grade, măsura unghiului M = 50 grade, masura unghiului N = 72 grade.
f) masura unghiului A = 81 grade, masura unghiului N = 67 grade, masura unghiului P = 32 grade, AB supra MN = AC supra MP.

Answer 1

   
$\displaystyle a) \\ AB=\frac{4MN}{5} ~~\Longrightarrow~~ \frac{AB}{MN}=\frac{4}{5} \\ \\ AC=0,8\cdot MP ~~\Longrightarrow~~\frac{AC}{MP}=0,8= \frac{8}{10} =\frac{4}{5} \\ \\ BC=80\%NP~~\Longrightarrow~~BC= \frac{80}{100} NP ~~\Longrightarrow~~ \frac{BC}{NP}= \frac{80}{100}=\frac{4}{5} \\ \\ \Longrightarrow~~ \text{Cazul de asemanare LLL }~~\Longrightarrow~~ \boxed{\Delta ABC \sim \Delta MNP}$


[tex]\displaystyle b)\\ \ \textless \ C = \ \textless \ P \\ \\ \frac{AB}{3} = \frac{MN}{4} ~~\Longrightarrow~~\frac{AB}{MN} = \frac{3}{4} \\ \\ \frac{BC}{21} = \frac{NP}{28} ~~\Longrightarrow~~ \frac{BC}{NP}= \frac{21}{28} = \frac{3}{4} \\ \\ \Longrightarrow~~ \text{Suntem in cazul prezentat in desenul atasat.} \\ \Longrightarrow~~ \boxed{\Delta ABC \sim \Delta MNP} [/tex]


[tex]\displaystyle c)\\ AB=6~cm\\BC=9~cm\\AC=12~cm\\MN=12~cm\\NP=9~cm\\MP=18~cm \\ \\ \frac{AB}{MN}= \frac{6}{12} = \frac{1}{2} \\ \\ \frac{BC}{NP}= \frac{9}{9} = 1 \\ \\ \frac{AC}{MP}= \frac{12}{18} = \frac{2}{3} \\ \\ \Longrightarrow ~~ \Delta ABC~~\texttt{ NU ESTE ASEMENEA CU }~~\Delta MNP [/tex]


[tex]\displaystyle d)\\ AB=1,2~dm \\ BC=1,6~dm \\ AC=1,8~dm \\ MN=0,6~dm \\ NP=0,8~dm \\ MP=0,9~dm \\ \\ \frac{AB}{MN}= \frac{1,2}{0,6}=2 \\ \\ \frac{BC}{NP}= \frac{1,6}{0,8}=2 \\ \\ \frac{AC}{MP}= \frac{1,8}{0,9}=2 \\ \\ \Longrightarrow ~~ \text{Suntem in cazul de asemanare LLL}\\ \Longrightarrow ~~ \boxed{\Delta ABC~\sim ~\Delta MNP} [/tex]


[tex]\displaystyle e)\\ \ \textless \ B =72^o \\ \ \textless \ C = 56^o \\ \Longrightarrow ~~\ \textless \ A = 180 -\ \textless \ B ~- \ \textless \ C =180-72-56 = 52^o\\ \\ \ \textless \ M = 50^o \\ \ \textless \ N = 72^o \\ \Longrightarrow ~~\ \textless \ P = 180 -\ \textless \ M ~- \ \textless \ N =180-50-72 = 58^o\\ \\ \text{Nu ne putem incadra in cazul LLL.} \\ \\ \Longrightarrow ~~ \Delta ABC~~\texttt{ NU ESTE ASEMENEA CU }~~\Delta MNP[/tex]


[tex]\displaystyle f)\\ \ \textless \ A = 81^o \\ \ \textless \ N = 67^o \\ \ \textless \ P = 32^o \\ \Longrightarrow ~~\ \textless \ M= 180~ -\ \textless \ N~ -\ \textless \ P = 180-67-32 = 81^o = \ \textless \ A\\ \frac{AB}{MN}= \frac{AC}{MP} \\ \\ \Longrightarrow ~~ \text{Suntem in cazul de asemanare LUL.} \\ \\ \Longrightarrow ~~ \boxed{\Delta ABC~\sim ~\Delta MNP}[/tex]