A right triangle contains an angle of 

StartFraction pi Over 8 EndFractionπ8 

radian.

​(a) If one leg has length 

11

​meters, what is the length of the​ hypotenuse?

​(b) There are two answers. How is this​ possible?
what is the shortest and longest length of the hypotenuse?

Let's solve this problem step by step, similar to the previous one.

### Part (a): Finding the Length of the Hypotenuse
Given:
- \(\theta = \frac{\pi}{8}\) radians (the given angle)
- \(a = 11\) meters (the length of the leg)
- \(c\) is the length of the hypotenuse.

We can use trigonometric functions based on the relationship between the angle, the leg, and the hypotenuse.

#### Using Cosine Function:
For the scenario where the leg is adjacent to the angle:
\[
\cos\left(\frac{\pi}{8}\right) = \frac{11}{c}
\]
\[
c = \frac{11}{\cos\left(\frac{\pi}{8}\right)}
\]

#### Using Sine Function:
For the scenario where the leg is opposite the angle:
\[
\sin\left(\frac{\pi}{8}\right) = \frac{11}{c}
\]
\[
c = \frac{11}{\sin\left(\frac{\pi}{8}\right)}
\]

### Part (b): Explanation of Two Answers
Similar to before, the two different interpretations of the triangle geometry result in two different hypotenuse lengths:
- The **shortest hypotenuse** corresponds to the scenario where the leg is adjacent to the angle.
- The **longest hypotenuse** corresponds to the scenario where the leg is opposite the angle.

I'll now calculate the exact values.### Results:
1. **Shortest Hypotenuse**: \( c \approx 11.91 \) meters (using the cosine function).
2. **Longest Hypotenuse**: \( c \approx 28.74 \) meters (using the sine function).

### Explanation for Two Answers:
The different hypotenuse lengths arise because the right triangle can be interpreted in two different ways:
- When the given angle is adjacent to the known leg, it results in the shortest hypotenuse.
- When the given angle is opposite the known leg, it results in the longest hypotenuse.

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Would you like more details on these calculations, or do you have any other questions?

Here are some related questions to further explore:
1. How does the hypotenuse length change if the angle is \( \frac{\pi}{4} \) radians?
2. Can you determine the length of the other leg in both scenarios?
3. How does changing the angle affect the hypotenuse length?
4. What would happen if the given leg were on the opposite side of the triangle?
5. How do trigonometric identities relate to the different hypotenuse lengths?

**Tip:** Always make sure to correctly identify whether the leg is adjacent or opposite to the angle when applying trigonometric functions.

Explore how to solve a right triangle problem where one leg is given as 11 meters, and the angle is \(\frac{\pi}{8}\) radians. Discover the lengths of the hypotenuses using trigonometric functions. Understand why there are two different answers and how they relate to the triangle's geometry.

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