Calculating Area of Square from Rectangle Diagonal

The problem asks us to find the area of a square whose side is equal to the length of the diagonal of a given rectangle. We are provided with the dimensions of the rectangle: length and breadth.

Let's first identify the given information:

• Length of the rectangle = 6 cm
• Breadth of the rectangle = 8 cm

Finding the Diagonal of the Rectangle

The diagonal of a rectangle forms a right-angled triangle with the length and breadth as the other two sides. We can use the Pythagorean theorem to find the length of the diagonal. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides (length and breadth).

Let $d$ be the length of the diagonal of the rectangle, $l$ be the length, and $b$ be the breadth.

According to the Pythagorean theorem:

$$d^2 = l^2 + b^2$$

Substituting the given values:

$$d^2 = (6 \text{ cm})^2 + (8 \text{ cm})^2$$

$$d^2 = 36 \text{ cm}^2 + 64 \text{ cm}^2$$

$$d^2 = 100 \text{ cm}^2$$

To find the length of the diagonal $d$, we take the square root of both sides:

$$d = \sqrt{100 \text{ cm}^2}$$

$$d = 10 \text{ cm}$$

So, the length of the diagonal of the rectangle is 10 cm.

Determining the Side of the Square

The problem states that the side of the square is equal to the length of the diagonal of this rectangle. Therefore,

Side of the square = Length of the diagonal = 10 cm.

Calculating the Area of the Square

The area of a square is calculated by squaring the length of its side. Let $s$ be the side of the square.

Area of the square = $s^2$

Substituting the side length we found:

Area of the square = $(10 \text{ cm})^2$$

Area of the square = $10 \times 10 \text{ cm}^2$$

Area of the square = $100 \text{ cm}^2$$

Thus, the area of the square is 100 cm$^2$.

Summary of Steps

The final calculated area of the square is 100 cm$^2$.

Revision Table: Rectangle and Square Properties

Additional Information: Pythagorean Theorem Explained

The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right-angled triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the other two sides (the legs).

If a right-angled triangle has sides of lengths $a$ and $b$ (the legs) and the hypotenuse has length $c$, the theorem can be written as:

$$a^2 + b^2 = c^2$$

In our problem, the length and breadth of the rectangle are the legs ($a=6$, $b=8$), and the diagonal is the hypotenuse ($c=d$). We used this theorem to find the length of the diagonal, which was then used as the side of the square to calculate its area.