Question

Aflati masurile unghiurilor unui patrulater convex ABCD in care masura unghiului A este media aritmetica a celorlalte trei masuri, masura unghiului B este media aritmetica a masurilor unghiurilor C si D, iar masura unghiului C este jumatate din masura unghiului D.

Answer 1

m∡A+m∡B+m∡C+m∡D=360°
NOTAM CU a=m∡A ; b=m∡B; c=m∡C ; d=m∡D
vom avea a+b+c+d=360
mai stim ca :
$a= \frac{b+c+d}{3}$
$b= \frac{c+d}{2}$
$c=\frac{d}{2}$
inlocuim si vom avea
$b= \frac{ \frac{d}{2} +d}{2}$
$b= \frac{\frac{d+2d}{2} } {2}$
$b= \frac{ \frac{3d}{2} }{2}$
$b= \frac{3d}{2} * \frac{1}{2} = \frac{3d}{4}$

$a= \frac{ \frac{3d}{4} + \frac{d}{2} +d}{3}$
$a= \frac{\frac{3d+2d+4d}{4} }{3}$
$a= \frac{ \frac{9d}{4} }{3} = \frac{9d}{4} * \frac{1}{3} = \frac{3d}{4}$

inlocuim in a+b+c+d=360 si vom avea
$\frac{3d}{4} + \frac{3d}{4} + \frac{d}{2} +d=360$
$\frac{3d+3d+2d+4d}{4} =360$
$\frac{12d}{4} =360$
3d=360
d=360:3
d=120 ⇒ m∡D=120°
$c= \frac{d}{2} = \frac{120}{2} =60$ ⇒m∡C=60°
$b= \frac{3d}{4} = \frac{3*120}{4} =3*30=90$ ⇒m∡B=90°
$b= \frac{3d}{4} = \frac{3*120}{4} =3*30=90$ ⇒m∡A=90°