Question

Calculate the area of a triangle / Найти площадь треугольника

Answer 1

Check the picture below.

so by the Inscribed Angle Theorem, we can see the arc made at "b" is 120°, so that means the central angle for that arc is 120°. Now let's use the Law of Cosines to find the base "b", keeping in mind the triangle has an altitude or height of 10.

`\textit{Law of Cosines}\\\\ c^2 = a^2+b^2-(2ab)\cos(C)\implies c = √(a^2+b^2-(2ab)\cos(C)) \\\\[-0.35em] ~\dotfill\\\\ b = \sqrt{(4√(3))^2+(4√(3))^2~-~2(4√(3))(4√(3))\cos(120^o)} \\\\\\ b = √( 48+48 - 96 \cos(120^o) ) \implies b = \sqrt{ 96 - 96(-(1)/(2)) }`

`b=√(96+48)\implies b=√(144)\implies b=12 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill~\stackrel{ \textit{\\ormalsize Area of the triangle} }{\cfrac{1}{2}(\underset{b}{12})(\underset{h}{10})\implies \text{\LARGE 60}}~\hfill~`