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
- The area of the greatest circle that can be inscribed inside a square of side 21cm is:
- Find the area of an equilateral triangle whose perimeter is 24 cm.
- The floor of a hall is rectangular. Its length is 30 m and breadth is 24 m. How many carpets of size 6m $\times$ 4m are required to cover it?
- If the difference between the exterior and the interior angles of a regular polygon is $60^{\circ}$, with an interior angle being greater than the corresponding exterior angle, then find the number of sides of the polygon.
- 4 angles of a pentagon are $70^\circ$, $110^\circ$, $135^\circ$ and $95^\circ$. Find the measure of the fifth angle of the pentagon.
- If the perimeter of a square is 20 cm, then what is its area?
- The sum of the interior angles of a polygon of 11 sides is:
- The length of each side of a regular hexagon is $2\sqrt{3} \text{ cm}$. What is the area of the given hexagon?
- The length of a rectangular plot is twice its breadth and the length of its diagonal is $6\sqrt{5}$ m. The perimeter of the plot is ______.
- The length and the breadth of a rectangular park are in the ratio 7 : 3. The perimeter of the park is 21000 m. What is the difference between the length and the breadth of the park?
Important Questions from Plane Figures
If length of a rectangle is increased to its three times and breadth is decreased to its half, then the ratio of the area of given rectangle to the area of new rectangle is:
The width of the path around a square field is 4.5 m and its area is 105.75 m 2. Find the cost of fencing the field at the rate of Rs. 100 per meter.
What is the area of the square (in cm 2) whose vertices lie on a circle of radius 5 cm?
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
The perimeter of a circular lawn is 1232 m. There is 7 m wide path around the lawn. The area (in m 2) of the path is:
Take \(\left(\pi=\frac{22}{7}\right)\)
