How to Unravel the Essential Properties of Rectangles

Step-by-step Guide: Properties of Rectangles

Definition of a Rectangle:
A rectangle is a four-sided polygon, known as a quadrilateral, wherein all four interior angles equal \(90^\circ\). This distinct feature ensures that opposite sides remain parallel to one another.

Equal Opposite Sides:
The hallmark of a rectangle is its equal opposite sides. If one side is \(a\) units long, its opposite counterpart will also be \(a\) units. Similarly, if an adjacent side measures \(b\) units, its opposite will measure the same, \(b\) units.

Area:
The area is a measure of the space enclosed by the rectangle. It is computed by multiplying its length by its breadth.
\(\text{Area} = a \times b\)

Perimeter:
The perimeter represents the total length of the rectangle’s boundaries. It’s calculated by adding all four sides together.
\(\text{Perimeter} = 2(a + b)\)

Diagonal:
Every rectangle boasts two equal-length diagonals. These diagonals not only bisect each other but also do so at right angles. To compute a diagonal’s length, we employ the Pythagorean theorem.
\(\text{Diagonal length} = \sqrt{a^2 + b^2}\)

Examples

Example 1:
Determining Area and Perimeter of a Rectangle.
Given: A rectangle with a length of \(10\) units and a breadth of \(5\) units.

Solution:
A rectangle’s area is calculated by multiplying its length by its breadth. On the other hand, its perimeter is obtained by summing up all its four sides. Since opposite sides of a rectangle are equal, the perimeter can be computed as twice the sum of its length and breadth.
Area:
\( \text{Area} = \text{Length} \times \text{Breadth} = 10 \times 5 = 50 \) square units
Perimeter:
\( \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) = 2(10 + 5) = 30 \) units

Example 2:
Computing the Diagonal of a Rectangle:
Given: A rectangle with a length of \(10\) units and a breadth of \(5\) units.

Solution:
The diagonal of a rectangle divides it into two right triangles. Using the Pythagoras theorem, where the diagonal acts as the hypotenuse, we can determine the diagonal’s length.
Diagonal:
\( \text{Diagonal} = \sqrt{\text{Length}^2 + \text{Breadth}^2} = \sqrt{10^2 + 5^2} = \sqrt{125} = 5\sqrt{5} \) units

Practice Questions:

1. Imagine a rectangle with a length of \(15\) units and a breadth of \(7\) units. Calculate its area, perimeter, and diagonal length.
2. A rectangle has a perimeter of \(44\) units. If its length exceeds its breadth by \(10\) units, determine the dimensions of this rectangle.
3. Rectangles have a cousin named the square. Given a square with a side measuring \(9\) units, compute its area, perimeter, and diagonal length.

Answers:

1. Area: \(15 \times 7 = 105\) square units. Perimeter: \(2(15 + 7) = 44\) units. Diagonal: \(\sqrt{15^2 + 7^2} = \sqrt{274} \approx 16.55\) units.
2. Let the breadth be (x). Thus, the length is \(x + 10\). From the given perimeter, \(2(x + x + 10) = 44\). Solving, we get \(x = 12\). So, the breadth is \(12\) units, and the length is \(22\) units.
3. Area: \(9 \times 9 = 81\) square units. Perimeter: \(4 \times 9 = 36\) units. Diagonal: \(\sqrt{9^2 + 9^2} = 9\sqrt{2} \approx 12.73\) units.