Answer:
[tex]\displaystyle ST ≈ 15,9\\ RT ≈ 30,8 \\ m∠S = 111°[/tex]
Step-by-step explanation:
First off, find [tex]\displaystyle m∠S:[/tex]
[tex]\displaystyle 180° = 58° + 31° + m∠S → 180° = 89° + m∠S \\ 111° = m∠S[/tex]
Now that the third angle has been defined, we can move move forward with solving for a second edge, using the Law of Sines, but before proceeding, here are some things you should know about this triangle:
- Edge s [RT] → [tex]\displaystyle m∠S[/tex]
- Edge t [RS] → [tex]\displaystyle m∠T[/tex]
- Edge r [ST] → [tex]\displaystyle m∠R[/tex]
Now that we have the information, we can proceed with the process:
[tex]\displaystyle \frac{t}{sin∠T} = \frac{s}{sin∠S} = \frac{r}{sin∠R} \\ \\ \frac{28}{sin\:58°} = \frac{s}{sin\:111°} → \frac{28sin\:111°}{sin\:58°} → 30,82402055... = s \\ 30,8 ≈ s[/tex]
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Now that RT has been defined, we can now proceed to solving the third edge, using the Law of Cosines:
[tex]\displaystyle s^2 + r^2 - 2sr\: cos∠T = t^2 \\ t^2 + r^2 - 2tr\: cos∠S = s^2 \\ t^2 + s^2 - 2ts\: cos∠R = r^2 \\ \\ 28^2 + 30,8^2 - 2[28][30]\: cos\:31° = r^2 → 784 + 948,64 - 1680\: cos\:31° = r^2 → \sqrt{254,1978397...} = \sqrt{r^2} → 15,94358303... = r \\ 15,9 ≈ r[/tex]
Your triangle is now complete!
I am joyous to assist you at any time.
