The area of a parallelogram is `180 cm^(2)`. If the ratio of its base and altitude is `9 : 5`, find the length of the base and corresponding altitude

Answer

To solve the problem, we need to find the length of the base and the corresponding altitude of a parallelogram given the area and the ratio of the base to the altitude.

### Step-by-Step Solution:

1. **Understand the given information:**
- Area of the parallelogram = 180 cm²
- Ratio of base to altitude = 9:5

2. **Let the base and altitude be expressed in terms of a variable:**
- Let the base = 9x
- Let the altitude = 5x
- This maintains the ratio of base to altitude as 9:5.

3. **Use the formula for the area of a parallelogram:**
- The area of a parallelogram is given by the formula:
\[ \text{Area} = \text{Base} \times \text{Altitude} \]
- Substituting the expressions for base and altitude, we have:
\[ 180 = (9x) \times (5x) \]

4. **Simplify the equation:**
- This simplifies to:
\[ 180 = 45x^2 \]

5. **Solve for x²:**
- Rearranging gives:
\[ x^2 = \frac{180}{45} \]
- Simplifying the right side:
\[ x^2 = 4 \]

6. **Find the value of x:**
- Taking the square root of both sides:
\[ x = 2 \]

7. **Calculate the base and altitude:**
- Now substitute x back to find the base and altitude:
- Base = \(9x = 9 \times 2 = 18 \, \text{cm}\)
- Altitude = \(5x = 5 \times 2 = 10 \, \text{cm}\)

8. **Final Answer:**
- The length of the base is **18 cm** and the corresponding altitude is **10 cm**.