Area Relationship: Triangle and Parallelogram on Same Base

This question relates to a fundamental theorem in geometry concerning the areas of figures sharing the same base and height.

Geometric Theorem

The theorem states that if a triangle and a parallelogram are situated on the same base and lie between the same parallels, then the area of the triangle is exactly half the area of the parallelogram.

Area Calculation

Let's denote:

• The common base as $b$.
• The common height (the perpendicular distance between the parallels) as $h$.

The formulas for the areas are:

• Area of the parallelogram: $ \text{Area}_\text{parallelogram} = \text{base} \times \text{height} = b \times h $
• Area of the triangle: $ \text{Area}_\text{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times b \times h $

Conclusion

Comparing the two formulas, we see that:

Therefore, the area of the triangle is half the area of the parallelogram.