The shortest side in a triangle with angles of 35, 65, and 80 degrees measures 16 m. Find the length of the longest side in the triangle.

Answer 1

Let's call the longest side of the triangle $x$. Since we know the measure of the smallest angle is opposite the smallest side, we can set up the following proportion: $$\frac{16}{\sin 35^\circ}=\frac{x}{\sin 80^\circ}.$$Now we can find $\sin 35$ and $\sin 80^\circ$. We have $\sin 35^\circ=\sin (45-10)^\circ = \sin 45^\circ \cos 10^\circ - \cos 45^\circ \sin 10^\circ = \frac{\sqrt{2}}{2}\left(\cos 10^\circ - \sin 10^\circ\right)$. We also have $\sin 80^\circ=\sin (45+35)^\circ=\sin 45^\circ \cos 35^\circ + \cos 45^\circ \sin 35^\circ = \frac{\sqrt{2}}{2}\left(\cos 35^\circ + \sin 35^\circ\right)$. Substituting these values into our proportion, we have \begin{align*}
\frac{16}{\frac{\sqrt{2}}{2}\left(\cos 10^\circ - \sin 10^\circ\right)}&=\frac{x}{\frac{\sqrt{2}}{2}\left(\cos 35^\circ + \sin 35^\circ\right)}\\
32&=\frac{x}{\frac{\sqrt{2}}{2}\left(\cos 35^\circ + \sin 35^\circ\right)}\\
x&=\boxed{64}.
\end{align*}

Answer 2

I can read what this says

Answer 3

Apologies for misunderstanding your input. In the given triangle, the shortest side has a length of 16 m. To find the longest side, we can make use of the Law of Sines. According to the Law of Sines, the ratio of the length of a side to the sine of the angle opposite that side is constant for all sides of a triangle. In this case, we can set up the following proportion:

$$\frac{16}{\sin 35^\circ}=\frac{x}{\sin 80^\circ}$$

To find the value of $\sin 35^\circ$ and $\sin 80^\circ$, we can use trigonometric tables or a calculator. Substituting these values into the proportion, we have:

$$\frac{16}{0.5736}=\frac{x}{0.1392}$$

Simplifying this equation, we find:

$$x=16 \times \frac{0.1392}{0.5736} \approx \boxed{3.876\, \text{m}}$$

Therefore, the length of the longest side in the triangle is approximately 3.876 m.

Answer 4

The longest side can not be shorter than the shortest side

Answer 5

You are correct, I apologize for the mistake in my previous response. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Since the shortest side has a length of 16 m, the longest side cannot be shorter than 16 m.

Therefore, it is not possible to determine the length of the longest side of the triangle with the given information.