Question

the diagonal of a rectangular field is 60meters more than the shorter Side. if the longer side is 30meters more than the shorter side, find the sides of the field​

Answer 1

Answer:

⋆ DIAGRAM :

$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\thicklines\put(7.7,3){\large{A}}\put(9.6,1.7){\sf{\large{(n + 60)}}}\put(7.7,1){\large{B}}\put(9.1,0.7){\sf{\large{(n + 30)}}}\put(11.1,1){\large{C}}\put(8,1){\line(1,0){3}}\put(8,1){\line(0,2){2}}\put(11,1){\line(0,3){2}}\put(8,3){\line(3,0){3}}\put(11.1,2){\sf{\large{n}}}\put(8,1){\line(3,2){3}}\put(11.1,3){\large{D}}\put(10.8,1){\line(0,2){0.2}}\put(10.8,1.2){\line(2,0){0.2}}\end{picture}$

$\rule{110}{0.8}$

• Let Shorter Side (Breadth) of Rectangular Field be n metre.
• Diagonal of Field = (n + 60) metre
• Length of Field = (n + 30) metre

⠀

✩ Property of Rectangle : Each Angle of Rectangle forms 90°, and so Form two Right Angle Triangle Inside it.

⠀

☢ $\underline{\textsf{By Pythagoras theorem in \triangle BCD :}}$

$:\implies\sf (Hypotenuse)^2=(Perpendicular)^2+(Base)^2\\\\\\:\implies\sf (n+60)^2=(n)^2+(n+30)^2\\\\\\:\implies\sf (n)^2 +(60)^2 + (2.n.60) = n^2 + (n)^2 + (30)^2 + (2.n.30)$

$\\:\implies\sf n^2 + 3600 + 120n = 2n^2 + 900 + 60n\\\\\\:\implies\sf 0 = 2n^2 - n^2 + 60n - 120n + 900 - 3600\\\\\\:\implies\sf n^2 - 60n - 2700 = 0\\\\\\:\implies\sf n^2 - (90 - 30)n - 2700 = 0\\\\\\:\implies\sf n^2 - 90n + 30n - 2700 = 0\\\\\\:\implies\sf n(n - 90) + 30(n - 90) = 0\\\\\\:\implies\sf (n - 90)(n + 30) = 0\\\\\\:\implies\green{\sf n = 90} \quad \sf or \quad \red{n = -\:30}$

⠀

✩ Ignoring Negative , value of n = 90

⠀⠀⠀$\rule{160}{1.5}$

$\bullet\:\:\textsf{Length = (n + 30) = \textbf{120 m}}\\\bullet\:\:\textsf{Breadth = n = \textbf{90 m}}\\\bullet\:\:\textsf{Diagonal = (n + 60) = \textbf{150 m}}$