In ΔSTU, *s* = 250 inches, m*m*∠U=17° and m*m*∠S=52°. Find the length of *t, to the nearest 10th of an inch.*

To find the length of side \( t \) in \( \Delta STU \), we can apply the **Law of Sines**. 

The Law of Sines states:

\[
\frac{s}{\sin(\angle U)} = \frac{t}{\sin(\angle S)} = \frac{u}{\sin(\angle T)}
\]

### Step 1: Find the measure of \( \angle T \).
Since the sum of the interior angles of a triangle is 180°, we can calculate \( \angle T \) as:

\[
\angle T = 180^\circ - \angle S - \angle U = 180^\circ - 52^\circ - 17^\circ = 111^\circ
\]

### Step 2: Apply the Law of Sines.
Now that we know all the angles, let’s plug the known values into the Law of Sines.

\[
\frac{250}{\sin(17^\circ)} = \frac{t}{\sin(52^\circ)}
\]

### Step 3: Solve for \( t \).
First, calculate the sines of the angles:

\[
\sin(17^\circ) \approx 0.2924, \quad \sin(52^\circ) \approx 0.7880
\]

Now, set up the equation:

\[
\frac{250}{0.2924} = \frac{t}{0.7880}
\]

\[
854.9 = \frac{t}{0.7880}
\]

Multiply both sides by 0.7880:

\[
t = 0.7880 \times 854.9 \approx 673.4 \, \text{inches}
\]

### Final Answer:
The length of side \( t \) is approximately **673.4 inches**.

---

Would you like further details or have any questions? 

Here are some related questions:  
1. What is the Law of Cosines, and when is it preferred over the Law of Sines?  
2. How do you solve a triangle when given two sides and a non-included angle (SSA case)?  
3. How does rounding intermediate values affect the final answer in trigonometry?  
4. What happens if the angles in a triangle are given in radians instead of degrees?  
5. How can you determine if a triangle is obtuse, acute, or right using side lengths?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) to avoid incorrect results in trigonometry!

In ΔSTU, s = 250 inches, m∠U = 17° and m∠S = 52°. Find the length of t, to the nearest 10th of an inch.

Learn how to find the length of side t in ΔSTU given s = 250 inches, ∠U = 17°, and ∠S = 52°. This problem uses trigonometry and the Law of Sines to calculate the unknown side, with a step-by-step solution. Suitable for Grades 10-12 students.

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