https://r2.erweima.ai/mathbot/image/1726191881958lnd1o56s.jpg

To solve this problem, we are given the perimeter of a rectangle and asked to find the length of side \( CD \).

### Step-by-step process:

1. **Perimeter of a rectangle formula**:
   The perimeter \( P \) of a rectangle is given by:
   \[
   P = 2 \times (\text{length} + \text{width})
   \]
   In this case, the perimeter is provided as 90 units.

2. **Sides of the rectangle**:
   From the image, we can see the expressions for the lengths of the sides:
   - Side \( CD = AB = 3z - 1 \)
   - Side \( AD = BC = 4z - 3 \)

3. **Substitute the expressions**:
   The formula for the perimeter becomes:
   \[
   90 = 2 \times \left( (3z - 1) + (4z - 3) \right)
   \]

4. **Simplify the equation**:
   \[
   90 = 2 \times (7z - 4)
   \]
   \[
   90 = 14z - 8
   \]

5. **Solve for \( z \)**:
   \[
   90 + 8 = 14z
   \]
   \[
   98 = 14z
   \]
   \[
   z = \frac{98}{14} = 7
   \]

6. **Find \( CD \)**:
   Now, substitute \( z = 7 \) into the expression for \( CD \):
   \[
   CD = 3z - 1 = 3(7) - 1 = 21 - 1 = 20
   \]

### Final answer:
\[
CD = 20 \text{ units}
\]

Would you like more details or have any questions about this solution?

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Here are 5 related questions to expand your understanding:
1. What would the perimeter be if \( z \) were equal to 8?
2. How do you derive the area of the rectangle based on the same side lengths?
3. What if the perimeter was 100 units instead of 90? What would \( z \) be?
4. How do different values of \( z \) affect the proportions of the rectangle?
5. What is the general formula for finding the sides of a rectangle if the perimeter is known?

**Tip**: Always simplify expressions inside parentheses first before multiplying!

El perímetro del rectángulo a continuación es 90 unidades. Hallar la longitud del lado CD.

Solve a problem where the perimeter of a rectangle is 90 units, and the task is to find the length of side CD using algebraic expressions. Learn step-by-step how to simplify and solve equations.

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