Parallelogram Area Ratio Theorem
The area of a parallelogram is calculated using the formula:
$ \text{Area} = \text{Base} \times \text{Height} $
Conditions for Equal Areas
Consider two parallelograms, Parallelogram 1 and Parallelogram 2.
- Equal Bases: The problem states that their bases are equal. Let Base$_1$ be the base of Parallelogram 1 and Base$_2$ be the base of Parallelogram 2. Then, $ \text{Base}_1 = \text{Base}_2 $
- Same Parallels: When two parallelograms are between the same parallels, they share the same perpendicular distance between these parallels. This distance is the height of both parallelograms. Let Height$_1$ be the height of Parallelogram 1 and Height$_2$ be the height of Parallelogram 2. Then, $ \text{Height}_1 = \text{Height}_2 $
Calculating the Area Ratio
Using the area formula:
- Area$_1 = \text{Base}_1 \times \text{Height}_1$
- Area$_2 = \text{Base}_2 \times \text{Height}_2$
Since $ \text{Base}_1 = \text{Base}_2 $ and $ \text{Height}_1 = \text{Height}_2 $, it follows that:
$ \text{Area}_1 = \text{Area}_2 $
The ratio of their areas is:
$ \frac{\text{Area}_1}{\text{Area}_2} = \frac{\text{Area}_1}{\text{Area}_1} = \frac{1}{1} $
Therefore, the ratio of the areas is $ 1 : 1 $.
Similar Questions
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Important Questions from Plane Figures
If the area of a square is 625 cm 2, then what is the perimeter of the square?
The area and the perimeter of a sheet of paper are 240 cm 2and 68 cm, respectively. What would be its length and breadth?
One side of rectangular field is 15 meters and one of its diagonals is 17 meters. Then find the area of the field.
The bisector of ∠B in ΔABC meets AC at D. If AB = 12 cm, BC = 18 cm and AC = 15 cm, then the length of AD (in cm) is:
The perimeter and the length of one of the diagonals of a rhombus is 26 cm and 5 cm respectively. Find the length of its other diagonal (in cm).
