The ratio of the areas of the triangle and a parallelogram in same parallels and on the same base is :

Options

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

Answer

To find the ratio of the areas of a triangle and a parallelogram that share the same base and are situated between the same parallel lines, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Shapes and Their Dimensions**:
- Let the base of both the triangle and the parallelogram be denoted as \( BC \).
- Let the height from the base to the top of the triangle and parallelogram be denoted as \( h \).

2. **Calculate the Area of the Parallelogram**:
- The formula for the area of a parallelogram is given by:
\[ \text{Area of Parallelogram} = \text{Base} \times \text{Height} \]
- Substituting the values, we have:
\[ \text{Area of Parallelogram} = BC \times h \]

3. **Calculate the Area of the Triangle**:
- The formula for the area of a triangle is given by:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- Substituting the values, we have:
\[ \text{Area of Triangle} = \frac{1}{2} \times BC \times h \]

4. **Set Up the Ratio of Areas**:
- Now, we need to find the ratio of the area of the triangle to the area of the parallelogram:
\[ \text{Ratio} = \frac{\text{Area of Triangle}}{\text{Area of Parallelogram}} = \frac{\frac{1}{2} \times BC \times h}{BC \times h} \]

5. **Simplify the Ratio**:
- Cancel out \( BC \) and \( h \) from the numerator and the denominator:
\[ \text{Ratio} = \frac{1/2}{1} = \frac{1}{2} \]

6. **Express the Ratio in Standard Form**:
- The ratio of the area of the triangle to the area of the parallelogram is:
\[ 1 : 2 \]

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