If l, b and p be the length, breadth and perimeter of a rectangle and b, l and p in GP (in order), then (l/b)² is:

Question

Question

Answer 1

Given:

l, b and p in GP where,

l = length, b = breadth and p = perimeter

Concept used:

(a - b)² = a² - 2ab + b²

(a + b)² = a² + 2ab + b²

Perimeter of rectangle = 2 × (l + b)

if a, b and c is in GP (in order) then,

b² = ac

Explanation:

According to the question

b, l, p are in G.P

So, l² = b × p

⇒ l² = b × 2(l + b)

⇒ l² - 2lb = 2b²

⇒ l² - 2lb + b² = 2b² + b²

⇒ (l - b)² = 3b²

⇒ l - b = √3b

⇒ l = √3b + b

⇒ l = b (√3 + 1)

⇒ l/b = (√3 + 1)/1

⇒ (l/b)² = (4 + 2√3)/1

∴ The ratio of l² : b² is (4 + 2√3) : 1.

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