Calculating the Area of the Rectangle

The problem involves a rhombus and a rectangle. We are given information about the rhombus and need to use it to find the dimensions and area of the rectangle.

Step 1: Find the Lengths of the Diagonals of the Rhombus

A rhombus has four equal sides, and its diagonals bisect each other at right angles. This property creates four congruent right-angled triangles inside the rhombus. The hypotenuse of each triangle is the side of the rhombus, and the legs are half the lengths of the diagonals.

• Side of the rhombus = 26 cm.
• Length of one diagonal (\(d_1\)) = 20 cm.
• Half of this diagonal = \(\frac{20}{2} = 10\) cm.

Let the other diagonal be \(d_2\). Half of this diagonal is \(\frac{d_2}{2}\) cm.

Using the Pythagorean theorem in one of the right-angled triangles:

$\$(\text{Side})^2 = (\frac{d_1}{2})^2 + (\frac{d_2}{2})^2\$\$$

Substituting the given values:

$\$26^2 = 10^2 + (\frac{d_2}{2})^2\$\$$

$\$676 = 100 + (\frac{d_2}{2})^2\$\$$

$\$(\frac{d_2}{2})^2 = 676 - 100\$\$$

$\$(\frac{d_2}{2})^2 = 576\$\$$

Taking the square root of both sides:

$\$ \frac{d_2}{2} = \sqrt{576} = 24 \text{ cm} \$\$$

So, the length of the other diagonal is:

$\$d_2 = 2 \times 24 = 48 \text{ cm}\$\$$

The lengths of the diagonals of the rhombus are 20 cm and 48 cm.

Step 2: Find the Sum of the Lengths of the Diagonals

The sum of the lengths of the diagonals is \(d_1 + d_2\).

$$ \text{Sum} = 20 \text{ cm} + 48 \text{ cm} = 68 \text{ cm} $$

Step 3: Relate Rhombus Diagonals to Rectangle Perimeter

The problem states that the sum of the lengths of the diagonals of the rhombus is equal to the perimeter of a rectangle.

Perimeter of rectangle = Sum of rhombus diagonals = 68 cm.

Let the length of the rectangle be \(l\) and the breadth be \(b\). The perimeter of a rectangle is given by the formula \(2(l + b)\).

$$ 2(l + b) = 68 \text{ cm} $$

Dividing both sides by 2:

$$ l + b = 34 \text{ cm} \quad \text{(Equation 1)} $$

Step 4: Use the Difference in Rectangle Dimensions

We are also given that the difference between the length and breadth of the rectangle is 6 cm.

$$ l - b = 6 \text{ cm} \quad \text{(Equation 2)} $$

Step 5: Solve for the Length and Breadth of the Rectangle

We have a system of two linear equations:

Equation 1: \(l + b = 34\)

Equation 2: \(l - b = 6\)

Adding Equation 1 and Equation 2:

$$ (l + b) + (l - b) = 34 + 6 $$

$$ 2l = 40 $$

$$ l = \frac{40}{2} = 20 \text{ cm} $$

Substitute the value of \(l\) into Equation 1:

$$ 20 + b = 34 $$

$$ b = 34 - 20 = 14 \text{ cm} $$

So, the length of the rectangle is 20 cm and the breadth is 14 cm.

Step 6: Calculate the Area of the Rectangle

The area of a rectangle is calculated by multiplying its length and breadth.

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

$$ \text{Area} = 20 \text{ cm} \times 14 \text{ cm} $$

$$ \text{Area} = 280 \text{ cm}^2 $$

The area of the rectangle is 280 cm\(^2\).