Question

If a perimeter of a rectangle is p and its diagonal is d then the difference between the lenght and width of the rectangle is

Answer 1

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Answer 2

Answer: $\sqrt{d^2+\frac{p^2}{4}-d^2}$

Step-by-step explanation:

Let l be the length of the rectangle and b be the width of the rectangle,

Then, the perimeter of the rectangle = 2 ( l + b )

And, the diagonal of the rectangle

$=\sqrt{l^2+b^2}$

According to the question,

2(l+b) = p ------ (1)

And,

$=\sqrt{l^2+b^2}=d$

⇒l² + b² = d² -----(2)

Squaring equation (1),

4(l+b)² = p²

⇒l² + b² + 2 lb = p²/ 4 -----(3)

Equation (3) - equation (2),

2 lb = p²/4 - d² ----- (4)

Thus, (l-b)² = l² + b² - 2lb = d² + p²/4 - d²

⇒ $l-b=\sqrt{d^2+\frac{p^2}{4}-d^2}$

Which is the required difference.