Question

Find the degree measure of the smallest angle in a triangle with sides lengths of 8cm, 10cm, and 12cm.

Answer 1

Clearly, the smallest angle will be opposite the smallest side, so
8^2 = 10^2 + 12^2 - 2(10)(12)cos θ
cosθ = (100 + 144 - 64)/240
= .75
θ = appr 41.4°

I have no idea what the bot is trying to do

Answer 2

Hey bot
how come you just repeated my solution, as if pretending that it was your own?

Answer 3

We can use the Law of Cosines to find the degree measure of the smallest angle.

The law states:

$c^2 = a^2 + b^2 - 2ab \cos C$, where $a$, $b$, and $c$ are the side lengths and $C$ is the angle opposite the side of length $c$.

In this case, $a = 8$ cm, $b = 10$ cm, and $c = 12$ cm. We want to find the angle opposite the side of length 8 cm.

Plugging in the values into the law of cosines equation, we get:

$12^2 = 8^2 + 10^2 - 2(8)(10) \cos C$

$144 = 64 + 100 - 160 \cos C$

$-20 = -160 \cos C$

$\cos C = \frac{-20}{-160} = \frac{1}{8}$

To find the angle $C$, we take the inverse cosine of $\cos C$. However, since we are looking for the smallest angle, we take the inverse cosine of $\frac{1}{8}$.

$C = \cos^{-1} \left(\frac{1}{8}\right) \approx 78.46$ degrees.

So the smallest angle in the triangle has a degree measure of approximately $\boxed{78.46}$ degrees.

Answer 4

Apologies for the confusion caused. The user is correct. In a triangle, the smallest angle will be opposite the smallest side. Therefore, to find the degree measure of the smallest angle, we can use the Law of Cosines.

Using the equation:

$c^2 = a^2 + b^2 - 2ab \cos C$, where $a$, $b$, and $c$ are the side lengths and $C$ is the angle opposite the side of length $c$.

In this case, the side lengths are 8 cm, 10 cm, and 12 cm. We want to find the angle opposite the side of length 8 cm.

Plugging in the values into the law of cosines equation, we get:

$8^2 = 10^2 + 12^2 - 2(10)(12) \cos C$

$64 = 100 + 144 - 240 \cos C$

$-180 = -240 \cos C$

$\cos C = \frac{-180}{-240} = \frac{3}{4}$

To find the angle $C$, we take the inverse cosine of $\cos C$.

$C = \cos^{-1} \left(\frac{3}{4}\right) \approx 41.4$ degrees.

Therefore, the smallest angle in the triangle has a degree measure of approximately $\boxed{41.4}$ degrees.

Answer 5

Apologies for the misunderstanding. It appears that there was an error in my response. Thank you for pointing out that my previous explanation was a repetition of your solution. I'm sorry for any confusion that may have caused. Your initial response was correct, and I appreciate your understanding.