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In this problem, we need to find the values of \( d \) (the diagonal of the rectangle in figure "a") and \( h \) (the height of the right triangle in figure "b") using the Pythagorean theorem.

### 1. Finding \( d \) in rectangle "a":
The Pythagorean theorem states that in a right triangle:
\[
c^2 = a^2 + b^2
\]
Where \( c \) is the hypotenuse (which corresponds to \( d \)), and \( a \) and \( b \) are the legs of the triangle.

For the rectangle, we can consider the diagonal as the hypotenuse and the sides of the rectangle as the legs. The sides of the rectangle are given as 2 cm and 4 cm.

So, we apply the Pythagorean theorem:
\[
d^2 = (4 \, \text{cm})^2 + (2 \, \text{cm})^2
\]
\[
d^2 = 16 + 4 = 20
\]
\[
d = \sqrt{20} \, \text{cm} \approx 4.47 \, \text{cm}
\]

### 2. Finding \( h \) in triangle "b":
In triangle "b", we know the base is 1 cm, and we are asked to find the height \( h \). We also know the hypotenuse (though not given explicitly, this is a right triangle).

We apply the Pythagorean theorem again:
\[
c^2 = h^2 + 1^2
\]
Assuming \( c = 2 \, \text{cm} \) (based on visual clues, since the figure may be scaled), we get:
\[
2^2 = h^2 + 1^2
\]
\[
4 = h^2 + 1
\]
\[
h^2 = 3
\]
\[
h = \sqrt{3} \, \text{cm} \approx 1.73 \, \text{cm}
\]

### Final values:
- \( d \approx 4.47 \, \text{cm} \)
- \( h \approx 1.73 \, \text{cm} \)

Would you like a more detailed explanation for any part of this process, or do you have any questions?

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Here are 5 related questions to deepen your understanding:
1. How would the value of \( d \) change if the dimensions of the rectangle were doubled?
2. Can the Pythagorean theorem be used to find the perimeter of the triangle? Why or why not?
3. How would you calculate the area of the rectangle and the triangle using the given information?
4. What is the relationship between the hypotenuse and the legs in a 45-45-90 triangle?
5. How would the Pythagorean theorem apply to a three-dimensional figure like a rectangular prism?

**Tip**: When solving geometry problems, always sketch the shapes and label the known values to help visualize the problem!

Encuentra el valor de las longitudes d y h en el rectángulo (a) y el triángulo (b), respectivamente, mostrados en la figura, utilizando el teorema de Pitágoras.

Learn how to find the lengths of a diagonal in a rectangle and the height of a right triangle using the Pythagorean theorem. This problem involves solving for unknown sides based on given dimensions and applying the theorem correctly. Suitable for middle school students.

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