Theorem 9.3: Area of a Parallelogram

The area of a parallelogram can be calculated by multiplying the length of its base by its corresponding height.

Explanation:

To understand Theorem 9.3, let's break it down into smaller sections.

Definition of a Parallelogram:
A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

Definition of Base and Height:
In a parallelogram, the base is any one of its parallel sides, and the height is the perpendicular distance between the base and its opposite side.

Formula for Calculating the Area:
The area of a parallelogram can be found using the formula:

Area = Base * Height

Illustration:
Consider a parallelogram with base length 'b' and corresponding height 'h':

- The base is represented by 'b' and is one of the parallel sides of the parallelogram.
- The height is represented by 'h' and is the perpendicular distance between the base and its opposite side.

Proof of Theorem 9.3:
To prove the theorem, we divide the parallelogram into small triangles and rearrange them to form a rectangle.

1. Divide the parallelogram into small triangles by drawing a line from one vertex to the opposite side parallel to the base.
2. Each triangle formed has a base equal to the length of the original base and a height equal to the height of the parallelogram.
3. Since the area of a triangle is given by (1/2) * base * height, the area of each small triangle is (1/2) * b * h.
4. There are two such triangles, so the combined area of both triangles is (2 * (1/2) * b * h) = b * h.
5. Rearrange the triangles to form a rectangle by joining their hypotenuses.
6. The rectangle formed has a length equal to the original base 'b' and a width equal to the height 'h'.
7. The area of a rectangle is given by length * width, which in this case is b * h.
8. Therefore, the area of the parallelogram is equal to the area of the rectangle, which is b * h.

Conclusion:
Theorem 9.3 states that the area of a parallelogram can be found by multiplying the length of its base by its corresponding height. This theorem is proven by dividing the parallelogram into small triangles and rearranging them to form a rectangle, which has the same area as the parallelogram.