A right triangle contains an angle of StartFraction pi Over 10 EndFraction
radian.
​(a) If one leg has length 5 ​meters, what is the length of the​ hypotenuse?
​(b) There are two answers. How is this​ possible?
what is the shortest and longest possible lengths of the hypotenuse?

Given the problem, let's tackle each part step by step:

### Part (a): Finding the Length of the Hypotenuse
In a right triangle, the relationship between the angle, one leg, and the hypotenuse can be described using trigonometric functions. Since we are given an angle and the length of one leg, we can use the cosine function.

Let:
- \(\theta = \frac{\pi}{10}\) radians (the given angle)
- \(a = 5\) meters (the length of the leg adjacent to the angle)
- \(c\) be the length of the hypotenuse.

The cosine function is defined as:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c}
\]

Substituting the known values:
\[
\cos\left(\frac{\pi}{10}\right) = \frac{5}{c}
\]

Now, solve for \(c\):
\[
c = \frac{5}{\cos\left(\frac{\pi}{10}\right)}
\]

### Part (b): Why Are There Two Answers?
The triangle's geometry can lead to different hypotenuse lengths depending on which side (adjacent or opposite) is associated with the given angle.

- **First Scenario**: When the given angle \(\frac{\pi}{10}\) is associated with the adjacent side (already calculated above).
- **Second Scenario**: When the given angle \(\frac{\pi}{10}\) is associated with the opposite side. In this case, we use the sine function.

For this scenario:
\[
\sin\left(\frac{\pi}{10}\right) = \frac{5}{c}
\]
\[
c = \frac{5}{\sin\left(\frac{\pi}{10}\right)}
\]

### Shortest and Longest Hypotenuse
The shortest possible hypotenuse will be the one associated with the cosine function (since cosine is greater than sine for the same angle), and the longest possible hypotenuse will be associated with the sine function.

Let’s calculate these:

1. **Shortest Hypotenuse**: \( c = \frac{5}{\cos\left(\frac{\pi}{10}\right)} \)
2. **Longest Hypotenuse**: \( c = \frac{5}{\sin\left(\frac{\pi}{10}\right)} \)

I'll now compute the exact values.### Results:
1. **Shortest Hypotenuse**: \( c \approx 5.26 \) meters (using the cosine function).
2. **Longest Hypotenuse**: \( c \approx 16.18 \) meters (using the sine function).

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

These two interpretations result in different hypotenuse lengths.

---

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

Here are some related questions to expand your understanding:
1. What would be the hypotenuse if the angle were \( \frac{\pi}{6} \) radians?
2. How would the problem change if the given leg were the opposite side?
3. What is the relationship between the angle and the length of the hypotenuse?
4. How can you find the length of the other leg in both scenarios?
5. How would the problem change if the leg length were doubled?

**Tip:** Always check whether an angle is adjacent or opposite to the given side when solving for the hypotenuse.

Learn how to calculate the hypotenuse of a right triangle given an angle of π/10 radians and one leg of length 5 meters. Understand why there are two possible lengths for the hypotenuse and how to determine the shortest and longest possible lengths. Suitable for high school students studying trigonometry.

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