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