Question

The sum of three sides of a rectangle = the sum of the fourth side and a diagonal. Prove that the ratio of the length : breadth = 1:√3​

Answer 1

Let ABCD is a rectangle

Let AB = CD = lLet BC = AD = b

In ∆ABC, we have

AC²=AB²+BC²   [Pythagoras theorem]

⇒AC² = l² + b²

⇒AC = √l²+b²

so, diagonal = √l²+b²

Now, sum of 3 sides = AB + BC + CD =l+b+l= 2l+b

Now, sum of fourth side and diagonal = AD + AC = b + √1²+b²

Now, b + √l²+b² = 2l + b

⇒l2+√b2 = 21

⇒l2+b2=4l²

⇒31² = b²

⇒√31 = b

⇒ $\frac{1}{b}$ = √$\frac{1}{3}$

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