Question

1.Patrulaterul convex ABCD are m( A) =m(C)=m(B)=m(D).Aflati m=(C).
2.Aflati masurile unghiurilor unui patrulater convex,stiind ca sunt direct proportionale cu numerele 11,15,21 si 25.
3.Aflati masurile unghiurilor unui patrulater convex ABCD stiind ca unghiurile A,B,C au masurile direct proportionale cu 4,5 si 6 iar unghiurile C si D au masurile direct proportionale cu 14 si 5.

Answer 1

[tex]1. \left \{ {{m(A)+m(B)+m(C)+m(D)=360} \atop {m( A) =m(C)=m(B)=m(D).}} \right. \\ =\ \textgreater \ 4m(C)=360=\ \textgreater \ m(C)=90 \\ 2.{m(A);m(B);m(C);m(D)] d.p. [11,15,21,25]} \\ \frac{m(A)}{11} = \frac{m(B)}{15} = \frac{m(C)}{21}= \frac{m(D)}{25} =k \\ m(A)=11k \\ m(B)=15k \\ m(C)=21k \\ m(D)=25k \\ m(A)+m(B)+m(C)+m(D)=360 \\ 11k+15k+21k+25k=360=\ \textgreater \ 72k=360=\ \textgreater \ k=5 \\ m(A)=11*5=55 \\ m(B)=15*5=75 \\ m(C)=21*5=105 \\ m(D)=25*5=125 \\ [/tex]
[tex]3. \frac{A}{4} = \frac{B}{5} = \frac{C}{6} \\ \frac{C}{14} = \frac{D}{5} \\ \frac{A}{4}= \frac{C}{6} =\ \textgreater \ A= \frac{4C}{6} \\ \frac{B}{5} = \frac{C}{6}=\ \textgreater \ B= \frac{5C}{6} \\ \frac{C}{14} = \frac{D}{5} =\ \textgreater \ C= \frac{5C}{14} \\ \frac{4C}{6}+\frac{5C}{6} +\frac{5C}{14}+C=360 \\ 14*4C+14*5C+6*5C+84C=360*84 \\ 56C+70C+30C+84C=360*84 \\ 240C=360*84 \\ C= \frac{360*84}{240} = \frac{3*84}{2} =3*42=126 \\ A= \frac{4*126}{6} = \frac{504}{6} =84 \\ B= \frac{5*126}{6} = \frac{630}{6} =105 \\ [/tex]
$D= \frac{5*126}{14} = \frac{630}{14} =45$