Calculating Rectangle Area Using Side and Diagonal

The problem asks us to find the area of a rectangle given the length of one side and the length of its diagonal. We are given that one side is 12 m and the diagonal is 13 m.

A rectangle has four interior angles, each measuring 90 degrees. When a diagonal is drawn across a rectangle, it divides the rectangle into two right-angled triangles. The sides of the rectangle form the legs of these right-angled triangles, and the diagonal acts as the hypotenuse.

In this specific problem, we have a right-angled triangle with one leg measuring 12 m (the given side of the rectangle) and the hypotenuse measuring 13 m (the diagonal). Let the other side of the rectangle (the unknown side, which is the other leg of the right-angled triangle) be denoted by $x$ meters.

We can use the Pythagorean theorem to find the length of the unknown side. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). Mathematically, this is represented as:

$\text{leg}_1^2 + \text{leg}_2^2 = \text{hypotenuse}^2$

Substituting the given values into the theorem, we get:

$12^2 + x^2 = 13^2$

Now, let's solve for $x$:

$144 + x^2 = 169$

Subtract 144 from both sides:

$x^2 = 169 - 144$

$x^2 = 25$

Take the square root of both sides:

$x = \sqrt{25}$

$x = 5$

So, the lengths of the sides of the rectangle are 12 m and 5 m.

The area of a rectangle is calculated by multiplying its length by its width:

Area = Length $\times$ Width

Area = 12 m $\times$ 5 m

Area = 60 square meters (sq. m)

Now let's compare our calculated area with the given options:

• Option 1: 60 sq.m
• Option 2: 55 sq.m
• Option 3: 50 sq.m
• Option 4: 45 sq.m

Our calculated area is 60 sq. m, which matches Option 1.

Therefore, the area of the rectangle is 60 sq.m.

Revision Table: Key Rectangle and Pythagorean Concepts

Additional Information on Rectangle Geometry

Understanding the properties of geometric shapes like rectangles is fundamental in geometry. The relationship between the sides and the diagonal through the Pythagorean theorem is a common application.

• Perimeter of a Rectangle: The total length of the boundary of the rectangle. Perimeter = 2 $\times$ (Length + Width). For our rectangle with sides 12 m and 5 m, the perimeter would be 2 $\times$ (12 + 5) = 2 $\times$ 17 = 34 m.
• Special Case - Square: A square is a special type of rectangle where all four sides are equal in length. If the sides are of length $s$, the area is $s^2$ and the diagonal is $\sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$.
• Units: It's important to pay attention to units. If the sides are in meters (m), the area is in square meters (sq. m or m$^2$).

Problems involving rectangles and diagonals often require the use of the Pythagorean theorem, making it a crucial tool in solving such geometry questions.