https://r2.erweima.ai/mathbot/image/1725925641105l7149bqf.jpg

The image shows two right triangles, and the problem asks to find the exact lengths of the sides labeled \(h\) and \(c\) in each triangle. It also specifies that the answers must be given in simplified radical form.

### First Triangle (on the left):
- This triangle is a 45°-45°-90° triangle with a hypotenuse of 5.
  
In a 45°-45°-90° triangle, the ratio of the sides is \(1 : 1 : \sqrt{2}\). This means that both legs are equal, and the hypotenuse is \( \text{leg} \times \sqrt{2} \).

1. Let the leg be \(h\). Since the hypotenuse is 5:
   \[
   h \times \sqrt{2} = 5 \quad \Rightarrow \quad h = \frac{5}{\sqrt{2}} = \frac{5 \sqrt{2}}{2}
   \]
   So, \(h = \frac{5 \sqrt{2}}{2}\).

### Second Triangle (on the right):
- This is a 30°-60°-90° triangle with the shorter leg (opposite the 30° angle) equal to 4.

In a 30°-60°-90° triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the shorter leg is opposite the 30° angle, the longer leg is opposite the 60° angle, and the hypotenuse is twice the shorter leg.

1. Let the hypotenuse be \(c\). Since the shorter leg is 4, the hypotenuse is:
   \[
   c = 2 \times 4 = 8
   \]

2. The longer leg (opposite the 60° angle) is:
   \[
   \text{longer leg} = 4 \times \sqrt{3} = 4\sqrt{3}
   \]

### Final Answers:
- For the first triangle: \( h = \frac{5\sqrt{2}}{2} \)
- For the second triangle: \( c = 8 \)

Let me know if you would like further clarification on any of the steps!

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Here are some questions to consider:
1. Why are the sides of a 45°-45°-90° triangle in the ratio \(1:1:\sqrt{2}\)?
2. What is the general formula for finding the hypotenuse in a 30°-60°-90° triangle?
3. How do we simplify radical expressions like \(\frac{5}{\sqrt{2}}\)?
4. Can you apply the same methods to find missing sides in other right triangles with different angles?
5. What are some real-world applications of 30°-60°-90° and 45°-45°-90° triangles?

**Tip:** When dealing with radical expressions, always rationalize the denominator to simplify the final answer!

En los siguientes triángulos rectángulos, hallar las longitudes exactas de los lados h y c. De ser necesario, escribir las respuestas en forma radical simplificada.

Learn how to find the exact lengths of sides in special right triangles (45°-45°-90° and 30°-60°-90°). This problem involves simplifying radical expressions and applying key trigonometric formulas.

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