Question

Considere um triângulo equilátero de lado 6cm e calcule o raio da circunferência circunscrita

Answer 1

$\text{Lei dos cossenos:}\\\\l^2=r^2+r^2-2\cdot{r}\cdot{r}\cdot\cos\alpha\\\\l^2=2r^2-2r^2\cdot\cos\alpha\\\\l^2=2r^2(1-\cos\alpha)\\\\\\\dfrac{l^2}{2(1-\cos\alpha)}=r^2\\\\\\r=\sqrt{\dfrac{l^2}{2(1-\cos\alpha)}}\\\\\\r=\sqrt{\dfrac{(6cm)^2}{2\left(1-\cos120^\circ\right)}}\\\\\\r=\sqrt{\dfrac{36cm^2}{2-2\cos120^\circ}}\\\\\\r=\sqrt{\dfrac{36cm^2}{2-2\left(-\dfrac{1}{2}\right)}}\\\\\\r=\sqrt{\dfrac{36cm^2}{2+2\cdot\dfrac{1}{2}}}\\\\\\r=\sqrt{\dfrac{36cm^2}{2+1}}\\\\\\r=\sqrt{\dfrac{36cm^2}{3}}\\\\\\r=\sqrt{12cm^2}\\\\\boxed{r=2\sqrt{3}\ cm^2}$