Answer
The dimensions of Carpet: are Length is\[12\text{ m}\], Diagonal is \[13\text{ m}\] and the Breadth is\[5\text{ m}\].
Work Step by Step
It is given that the breadth of a rectangular carpet is 7 m less than its length and the diagonal is 1 m more than its length.
Let length of rectangular carpet be\[l\]. Therefore, its breadth is \[\left( l-7 \right)\]m and diagonal be \[\left( l+1 \right)\]m.
In rectangle ABCD, AC is the diagonal, AB and CD are the lengths and AD and BC are the breadths.
\[\Delta ABC\]is a right triangle right angled at angle B as ABCD is rectangle. Therefore, all the angles intersect each other at right angle.
Compute value of l by using Pythagorean Theorem in \[\Delta ABC\] and substitute the value of a and b into\[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\].
\[\begin{align}
& {{c}^{2}}={{a}^{2}}+{{b}^{2}} \\
& A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}} \\
& {{\left( l+1 \right)}^{2}}={{\left( l \right)}^{2}}+{{\left( l-7 \right)}^{2}} \\
& {{l}^{2}}+2l+1={{l}^{2}}+{{l}^{2}}-14l+49
\end{align}\]
Re arranging the equation as follows:
\[\begin{align}
& 2l+1={{l}^{2}}-14l+49 \\
& 0={{l}^{2}}-14l+49-2l-1 \\
& 0={{l}^{2}}-16l+48 \\
& 0={{l}^{2}}-12l-4l+48
\end{align}\]
Solving further the equation as follows;
\[\begin{align}
& 0=l\left( l-12 \right)-4\left( l-12 \right) \\
& 0=\left( l-4 \right)\left( l-12 \right) \\
& l=4,12
\end{align}\]
Hence, the value of l is\[12\]. Compute the value of diagonal and breadth as shown below:
\[\begin{align}
& \text{Diagonal}=\left( l+1 \right) \\
& =12\text{ m}+1\text{ m} \\
& =13\text{ m}
\end{align}\].
\[\begin{align}
& \text{Breadth}=\left( l-7 \right) \\
& =12\text{ m}-7\text{ m} \\
& =5\text{ m}
\end{align}\]
Hence, the dimensions of Carpet are Length is\[12\text{ m}\], Diagonal is \[13\text{ m}\] and the Breadth is\[5\text{ m}\].
