Question

Instead of walking along two adjacent sides of a rectangular field a man took a shortcut along the diagonal of the field and saved the distance of half the longer side then the ratio of shorter to the longer side is? ​

Answer 1

Let the Rectangular Field be represented by : ABCD

Let the Length of the Rectangular Field (AB) be : L

Let the Width of the Rectangular Field (BC) be : W

Goal of the Man : To reach the point C

The Man can reach the point C in two ways. They are :

★  Walk along two adjacent sides AB (Length) and BC (Width)

★  Walk along the Diagonal of the field (AC)

Case - 1 : Man walking along two adjacent sides AB and BC

In this case, The Total Distance covered by the Man to reach Point C will be Sum of Lengths of the Sides AB and BC

✪  Total Distance covered by the Man = L + W

Case - 2 : Man walking along the diagonal of the field (AC)

In this case, The Total Distance covered by the Man to reach Point C will be Length of the Diagonal AC

We know that : Diagonal of a Rectangle is the Hypotenuse of the Right angled Triangle formed by the Diagonal and the Adjacent Sides (legs)

From Pythagorean Theorem, We know that :

★  (Hypotenuse)² = (First Leg)² + (Second Leg)²

$\implies$  (Length of the Diagonal)² = (Length)² + (Width)²

$\implies$  (AC)² = (L)² + (W)²

$\mathsf{\implies AC = \sqrt{L^2 + W^2}}$

Given : Instead of Walking along two adjacent sides of the rectangular field the man took the shortcut along the diagonal of the field and saved the distance of half of the longer side

It means : if he walks along the Diagonal of the field, He saves a distance of half of the longer side (Length of the rectangular field)

It also means : The Difference between the lengths which he walked in the Two cases which are mentioned above should be equal to half of the Length of the rectangular field.

$\mathsf{\implies L + W - \sqrt{L^2 + W^2} = \dfrac{L}{2}}$

$\mathsf{\implies \dfrac{L}{2} + W = \sqrt{L^2 + W^2}}$

$\textsf{Squaring on both sides, We get :}$

$\mathsf{\implies \bigg(\dfrac{L}{2} + W\bigg)^2 = L^2 + W^2}$

$\mathsf{\implies \dfrac{L^2}{4} + W^2 + LW = L^2 + W^2}$

$\mathsf{\implies LW = L^2 - \dfrac{L^2}{4}}$

$\mathsf{\implies LW = \dfrac{4L^2 - L^2}{4}}$

$\mathsf{\implies LW = \dfrac{3L^2}{4}}$

$\mathsf{\implies W = \dfrac{3L}{4}}$

$\mathsf{\implies \dfrac{W}{L} = \dfrac{3}{4}}$

$\textsf{We know that : Width is the Shorter Side and Length is the Longer Side}$

$\underline{\bf{Answer}} \implies \textsf{The Ratio of Shorter to Longer Side is 3 : 4}$

Answer 2

Longer Side=Length
Shorter Side=Breadth

Let Length be=X
Let Breadth be=y

Diagonal of Rectangle=$\sqrt{{x}^{2}+{y}^{2}}$

If Person Take Diagonal Path he will Cover Distance upto Half of Length till reaching a point.

Diagonal=$\frac{x}{2}+y$

A/Q
$\sqrt{{x}^{2}+{y}^{2}}=\frac{x}{2}+y$

Transferring Square Root to RHS become

${x}^{2} + {y}^{2} = ( \frac{x}{2} + y) {}^{2}$
We know (a+b)²=a²+b²+2ab
Now
${x}^{2} + {y}^{2} = \frac{ {x}^{2} }{4} + {y}^{2} + \cancel{2} \times \frac{x}{\cancel{2} } \times y \\ \\ {x}^{2} + \cancel {y}^{2} - \frac{ {x}^{2} }{4} -\cancel {y}^{2} = xy \\ \\ xy = \frac{4 {x}^{2} - {x}^{2} }{4} \\ \\ xy = \frac{3 {x}^{2} }{4} \\ \\ y = \frac{ {3 \cancel { x}^{2}} }{4 \: \cancel x} \\ \\ y = \frac{3x}{4} \\ \\ \frac{y}{x} = \frac{3}{4} \\ \\ y : x = 3 : 4$
Breadth: Length=3:4

$\boxed{\mathbf{Ratio=3:4}}$