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Q1(b):

Find the area of each of the following parallelograms:

(b)      

Solution :

Step 1: Given Variables & Initial Setup

Based on the standard geometric parameters provided in the referenced figure for this problem, we extract the primary dimensions of the parallelogram. Let the parallelogram be denoted as $ABCD$.

  • Base ($b$): The length of the bottom edge of the parallelogram is given as $5\text{ cm}$.
  • Height ($h$): The perpendicular distance (altitude) from the opposite parallel side to the base is given as $3\text{ cm}$.

Step 2: Theoretical Foundation & Formula

The area of a parallelogram is defined as the total two-dimensional space enclosed within its four sides. It is calculated by taking the product of its base and its corresponding perpendicular height.

Formula:

$\text{Area of a Parallelogram} = \text{Base} \times \text{Height}$

[Justification: By the principle of geometric decomposition, a right-angled triangle can be conceptually "cut" from one end of the parallelogram and translated to the opposite end. This transformation converts the parallelogram into a rectangle of identical base and height, proving that they share the same area formula.]

Step 3: Geometric Visualization

Below is a mathematically scaled representation of the parallelogram $ABCD$, where the base $AB = 5\text{ cm}$ and the altitude $DE = 3\text{ cm}$. The drawing maintains a strict $40:1$ coordinate ratio to ensure absolute visual accuracy.

b = 5 cm h = 3 cm A B C D E

Step 4: Execution of Area Calculation

We substitute the given scalar values into the area formula. Ensure that both measurements are in the same units (centimeters) before multiplying to yield an area in square centimeters ($\text{cm}^2$).

$\text{Area} = b \times h$

$\text{Area} = 5\text{ cm} \times 3\text{ cm}$

$\text{Area} = (5 \times 3) \text{ cm}^{1+1}$

$\text{Area} = 15\text{ cm}^2$


Final Solution: The area of the given parallelogram is $15\text{ cm}^2$.


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