Răspuns:
ex.16,17
Explicație pas cu pas:
16. notăm:
[tex]\widehat {BC} = x[/tex]
atunci:
[tex]\widehat {AC} = 2 \cdot \widehat {BC} = 2x[/tex]
și
[tex]\widehat {AB} = 3 \cdot \widehat {AC} = 3 \cdot 2x = 6x[/tex]
[tex]\widehat {AB} + \widehat {AC} + \widehat {BC} = 360 \degree \iff 6x + 2x + x = 360 \degree \\ 9x = 360 \degree \implies x = 40 \degree[/tex]
=>
[tex]\widehat {AB} = 240 \degree \implies \measuredangle ACB = \frac{1}{2} \cdot \widehat {AB} \\ \implies \bf \measuredangle ACB = 120 \degree\\ \widehat {AC} = 80 \degree \implies \measuredangle ABC = \frac{1}{2} \cdot \widehat {AC} \\ \implies \bf \measuredangle ABC = 40 \degree\\ \widehat {BC} = 40 \degree \implies \measuredangle BAC = \frac{1}{2} \cdot \widehat {BC} \\ \implies \bf \measuredangle BAC = 20 \degree[/tex]
.
17. notăm:
[tex]\widehat {BC} = x[/tex]
atunci:
[tex]\widehat {BC} = 0,5 \cdot \widehat {AB} \implies \widehat {AB} = 2x[/tex]
[tex]3 \cdot \widehat {AB} = 2 \cdot \widehat {AC} \implies 3 \cdot 2x = 2 \cdot \widehat {AC} \\ \iff \widehat {AC} = 3x[/tex]
[tex]\widehat {AB} + \widehat {AC} + \widehat {BC} = 360 \degree \iff 2x + 3x + x = 360 \degree \\ 6x = 360 \degree \implies x = 60 \degree[/tex]
=>
[tex]\widehat {AB} = 2x = 120 \degree \implies \measuredangle ACB = \frac{1}{2} \cdot \widehat {AB} \\ \implies \bf \measuredangle ACB = 60 \degree\\ \widehat {AC} = 3x = 180 \degree \implies \measuredangle ABC = \frac{1}{2} \cdot \widehat {AC} \\ \implies \bf \measuredangle ABC = 90 \degree\\ \widehat {BC} = x = 60 \degree \implies \measuredangle BAC = \frac{1}{2} \cdot \widehat {BC} \\ \implies \bf \measuredangle BAC = 30 \degree[/tex]
