If a triangle and a parallelogram are on the same base and between same parallels, then the ratio of the area of the triangle to the area of parallelogram is

Options

A. `1 : 3`
B. `1 : 2`
C. `3 :1`
D. `1 : 4`

Explanation

To solve the problem, we need to establish the relationship between the areas of a triangle and a parallelogram that share the same base and are situated between the same parallels.

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have a triangle and a parallelogram.
- Both shapes share the same base and are located between the same set of parallel lines.

2. **Area of Parallelogram**:
- The area of a parallelogram is calculated using the formula:
\[ \text{Area of Parallelogram} = \text{Base} \times \text{Height} \]

3. **Area of Triangle**:
- The area of a triangle is calculated using the formula:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

4. **Establishing the Relationship**:
- Since both shapes share the same base and height (the height is the perpendicular distance between the two parallel lines), we can express the area of the triangle in terms of the area of the parallelogram:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Area of Parallelogram} \]

5. **Finding the Ratio**:
- To find the ratio of the area of the triangle to the area of the parallelogram, we set up the following equation:
\[ \frac{\text{Area of Triangle}}{\text{Area of Parallelogram}} = \frac{\frac{1}{2} \times \text{Area of Parallelogram}}{\text{Area of Parallelogram}} \]
- Simplifying this gives:
\[ \frac{\text{Area of Triangle}}{\text{Area of Parallelogram}} = \frac{1}{2} \]

6. **Expressing the Ratio**:
- The ratio of the area of the triangle to the area of the parallelogram can be expressed as:
\[ 1 : 2 \]

### Conclusion:
The ratio of the area of the triangle to the area of the parallelogram is \(1 : 2\).