In ΔSTU, *u* = 52 inches, *s* = 45 inches and ∠T=23°. Find the length of *t*, *to the nearest inch*.

To find the length of side \( t \) in \(\triangle STU\), given:

- \( u = 52 \) inches
- \( s = 45 \) inches
- \( \angle T = 23^\circ \)

we can use the **Law of Cosines**, which states:
\[
t^2 = s^2 + u^2 - 2 \cdot s \cdot u \cdot \cos(\angle T)
\]

### Step-by-Step Solution

1. Substitute the given values:
   \[
   t^2 = 45^2 + 52^2 - 2 \cdot 45 \cdot 52 \cdot \cos(23^\circ)
   \]

2. Calculate each part:
   - \( 45^2 = 2025 \)
   - \( 52^2 = 2704 \)
   - \( 2 \cdot 45 \cdot 52 = 4680 \)

3. Substitute and compute:
   \[
   t^2 = 2025 + 2704 - 4680 \cdot \cos(23^\circ)
   \]

4. Using \(\cos(23^\circ) \approx 0.9205\):
   \[
   t^2 = 2025 + 2704 - 4680 \cdot 0.9205
   \]
   \[
   t^2 = 2025 + 2704 - 4300.74
   \]
   \[
   t^2 \approx 428.26
   \]

5. Take the square root to find \( t \):
   \[
   t \approx \sqrt{428.26} \approx 21
   \]

### Answer:
The length of \( t \) is approximately **21 inches**.

Would you like further details or have any questions?

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Here are some related questions for further exploration:

1. How can the Law of Sines be applied to this triangle?
2. If the angle opposite side \( s \) was given, could we find the area of \(\triangle STU\)?
3. How does changing \(\angle T\) affect the length of \( t \)?
4. How would this problem differ if it was a right triangle?
5. Can we determine any other angle in \(\triangle STU\) with only one angle given?

**Tip:** When working with the Law of Cosines, remember it’s helpful for non-right triangles, especially when two sides and the included angle are known.

In ΔSTU, u = 52 inches, s = 45 inches and ∠T = 23°. Find the length of t, to the nearest inch.

Calculate the length of side t in ΔSTU, given u = 52 inches, s = 45 inches, and ∠T = 23°. This problem demonstrates the Law of Cosines, a key trigonometric concept for non-right triangles. Suitable for high school students, Grades 10-12.

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