Questions: Find the area of the following triangle: Not Enough Information 2.9 units squared 3.9 units squared 4.9 units squared 5.9 units squared

Solution Steps

Let the sides be $a=2$, $b=3$, and the angle between them be $C = 75^{\circ}$. We can use the Law of Cosines to find the third side, $c$.

$c^2 = a^2 + b^2 - 2ab \cos C$ $c^2 = 2^2 + 3^2 - 2(2)(3) \cos 75^{\circ}$ $c^2 = 4 + 9 - 12 \cos 75^{\circ}$ $c^2 = 13 - 12(0.2588)$ $c^2 \approx 13 - 3.1056$ $c^2 \approx 9.8944$ $c \approx \sqrt{9.8944} \approx 3.1455$

Let $s$ be the semiperimeter of the triangle: $s = \frac{a+b+c}{2} = \frac{2+3+3.1455}{2} \approx \frac{8.1455}{2} \approx 4.07275$

The area of the triangle is given by Heron's formula: $Area = \sqrt{s(s-a)(s-b)(s-c)}$ $Area \approx \sqrt{4.07275(4.07275-2)(4.07275-3)(4.07275-3.1455)}$ $Area \approx \sqrt{4.07275(2.07275)(1.07275)(0.92725)}$ $Area \approx \sqrt{8.6934}$ $Area \approx 2.95$

Final Answer