To find the dimensions of the rectangle, let's use algebraic methods. We'll designate:
- The length of the rectangle as \(\( L \)\) (in metres)
- The breadth of the rectangle as \(\( B \)\) (in metres)
We are given the following information:
1. The area of the rectangle is \(\( 8 \)\) square metres: \(\( A = L \times B = 8 \)\)
2. The breadth is two metres less than the length: \(\( B = L - 2 \)\)
Now let's use the given information to set up equations:
Firstly, we have \(\( A = L \times B \)\), and since the area \(\( A \)\) is given as \(\( 8 \)\), we have:
\(\[ L \times B = 8 \quad \text{(Equation 1)} \]\)
Secondly, the relationship between the breadth and length is \(\( B = L - 2 \)\). We can express \(\( B \)\) in terms of \(\( L \)\) to use it in our first equation:
\(\[ B = L - 2 \quad \text{(Equation 2)} \]\)
Substituting Equation 2 into Equation 1, we get an equation solely in terms of \(\( L \)\):
\(\[ L \times (L - 2) = 8 \]\)
Expanding this, we get:
\(\[ L^2 - 2L = 8 \]\)
Rearranging the terms to set the quadratic equation to zero, we have:
\(\[ L^2 - 2L - 8 = 0 \]\)
This is a quadratic equation in standard form. We can solve it using factoring, completing the square, or using the quadratic formula. We'll use the factoring method here:
We want two numbers that multiply to \(\( -8 \)\) and add to \(\( -2 \)\). Those numbers are \(\( -4 \)\) and \(\( 2 \)\). So we can factor our equation as:
\(\[ (L - 4)(L + 2) = 0 \]\)
Setting each factor equal to zero gives us two possible solutions for \(\( L \)\):
\(\[ L - 4 = 0 \quad \text{or} \quad L + 2 = 0 \]\)
Solving each equation for \(\( L \)\) yields:
\(\[ L = 4 \quad \text{or} \quad L = -2 \]\)
Since a length cannot be negative, we discard \(\( L = -2 \)\). Thus, the length of the rectangle is \(\( 4 \)\) metres.
Now that we know the length \(\( L \)\), we can find the breadth \(\( B \)\) using Equation 2:
\(\[ B = L - 2 = 4 - 2 = 2 \]\)
So, the breadth of the rectangle is \(\( 2 \)\) metres.
In conclusion, the dimensions of the rectangle are \(\( 4 \)\) metres in length and \(\( 2 \)\) metres in breadth.
