Question

3. Pe un cerc se consideră punctele A, B și C. Aflaţi măsurile unghiurilor triunghiului ABC, ştiind că măsurile arcelor AB, BC și CA sunt direct proporţionale cu numerele 2, 3 si 4.
ESTE URGENT !!!
DAU COROANĂ

Answer 1

Răspuns:

$\boxed{\mathbf{\angle A= 40^{\circ}}}$

$\boxed{\mathbf{\angle B=60^{\circ}}}$

$\boxed{\mathbf{\angle C = 80^{\circ}}}$

Explicație pas cu pas:

$\mathbf{A,B,C \in C(O;r)}$

$\mathbf{\bigg(\mathbf{\widehat{AB}; \widehat{BC}; \widehat{CA} \bigg)} d.p.(2;3;4)}$

$\mathbf{\implies \dfrac{\widehat{AB}}{2}=\dfrac{\widehat{BC}}{3}=\dfrac{\widehat{CA}}{4} =k }$

$\mathbf{\widehat{AB}=2k}$

$\mathbf{\widehat{BC}=3k}$

$\mathbf{\widehat{CA}=4k}$

$\mathbf{m(\widehat{AB})+(\widehat{BC})+m(\widehat{CA})=m(C(O;r))=360^{\circ}}$

$\mathbf{\implies 2k+3k+4k=360^{\circ}}$

$\mathbf{\implies 9k=360^{\circ}\; |:9 \implies k=40^{\circ}}$

$\mathbf{\widehat{AB}=2k=2 \cdot 40^{\circ}=80^{\circ}}$

$\mathbf{\widehat{BC}=3k=3 \cdot 40^{\circ}=120^{\circ}}$

$\mathbf{\widehat{CA}=4k=4 \cdot 40^{\circ}=160^{\circ}}$

$\mathbf{\implies \widehat{AB};\, \widehat{BC} ;\, \widehat{CA}-unghiuri \; inscrise\: in \ cerc}$

$\mathbf{\implies \angle A=\dfrac{\widehat{BC}}{2} =\dfrac{80^{\circ}}{2} =40^{\circ}}$

$\mathbf{\implies \angle B=\dfrac{\widehat{AC}}{2}=\dfrac{120^{\circ}}{2}=60^{\circ} }$

$\mathbf{\implies \angle C=\dfrac{\widehat{AB}}{2} =\dfrac{160^{\circ}}{2} =80^{\circ}}$

$\star$

Bafta! :)

#copaceibrainly