14682

14682 (Cristina Vijdeluc și Mihai Vijdeluc)

Enunț: Se consideră triunghiul ABC în care Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(\angle A) = 2 \cdot m(\angle B) + 30^\circ} . Punctul M este situat pe segmentul (BC) astfel încât AM = AC.

Dacă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(\angle MAC) = 2 \cdot m(\angle MAB)} , arătați că BM = MC.

Soluție: Notăm Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = m(\angle ABC)} și $x=m(\angle BAM)$. Avem $m(\angle BAC)=2a+30^{\circ }$ și $m(\angle CAM)=2x$, din ipoteză. Atunci $3x=2a+30^{\circ }$ de unde $x={\frac {2a}{3}}+10^{\circ }$. Pe de altă parte avem $m=(\angle AMC)=a+x={\frac {5a}{3}}+10^{\circ }$ ca unghi exterior $\triangle AMB$. Cum AM = AC vom avea Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(\angle ACM) = \frac {5a}{3} + 10^\circ} . Acum în Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \triangle ABC} avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + 2a + 30^\circ + \frac{5a}{3} + 10^\circ = 180^\circ} , de unde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 30^\circ} , apoi Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 30^\circ} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(\angle AMC) = 60^\circ} . Rezultă acum că triunghiul ABM este isoscel, de unde BM = AM, iar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \triangle AMC} (1) este echilateral AM = AC = CM,(2). Din (1) și (2) rezultă BM = MC.