Question

a triangle has sides of length 11 m, 12 m, and 16 m. what is the measure of the angle opposite the side that is 16 m long? round to the nearest degree. \\(20^{\circ}\\) \\(34^{\circ}\\) \\(74^{\circ}\\) \\(88^{\circ}\\)

Explanation:

Step1: Apply Law of Cosines

Let the angle opposite 16 m be $\theta$. The Law of Cosines states:
$$\cos\theta = \frac{a^2 + b^2 - c^2}{2ab}$$
where $a=11$, $b=12$, $c=16$.

Step2: Substitute values

$$\cos\theta = \frac{11^2 + 12^2 - 16^2}{2\times11\times12}$$
$$\cos\theta = \frac{121 + 144 - 256}{264}$$
$$\cos\theta = \frac{9}{264} \approx 0.0341$$

Step3: Calculate inverse cosine

$$\theta = \cos^{-1}(0.0341) \approx 88^\circ$$

Answer:

88°