Question

length and breadth of a rectangle are in ratio 4:3 and 3 equilateral triangles are constructed in such a way that one is on length and one is on breadth and one is on the diagonal find the ratio of ther area of triangles
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Answer 1

Given :-

• Ratio of Length & Breadth = 4:3
• 3 Equaliteral ∆ are constructed on Length , Breadth & Diagonal of Square .

To Find :-

• Ratio of Area of Constructed 3 ∆'s.

Formula used :-

• Diagonal of Rectangle = √(L² + B²)
• Area of Equaliteral ∆ = (√3/4) * (side)²

Solution :-

Let us Assume That, Length & Breadth of Rectangle are 4x & 3x respectively.

So,

→ Length = 4x

→ Each side of ∆ constructed on Length = 4x

→ Area of Equaliteral ∆ so formed = (√3/4) * (4x)² = 4√3x² -------------- Equation (1) .

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Similarly,

→ Breadth = 3x

→ Each side of ∆ constructed on Breadth = 3x

→ Area of Equaliteral ∆ so formed = (√3/4) * (3x)² = (9√3x²)/4 -------------- Equation (2) .

______________

Now,

→ Diagonal of Rectangle = √(L² + B²)

→ Diagonal = √[(4x)² + (3x)²] = √(16x² + 9x²) = √(25x²) = 5x

So,

→ Area of Equaliteral ∆ so formed on Diagonal = (√3/4) * (5x)² = (25√3x²)/4 -------------- Equation (3) .

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From Equation (1) , (2) & (3) , we get :-

→ Ratio of Area of ∆₁ : ∆₂ : ∆₃ = (4√3x²) : [(9√3x²)/4 : (25√3x²)/4

→ ∆₁ : ∆₂ : ∆₃ = (4√3x²) : [(9√3x²)/4 : (25√3x²)/4

→ ∆₁ : ∆₂ : ∆₃ = [(16√3x²)/4] : [(9√3x²)/4 : (25√3x²)/4

Now, (1/4) * (√3x²) will cancel from All ,

So ,

→ ∆₁ : ∆₂ : ∆₃ = 16 : 9 : 25 (Ans.)

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Shortcut :-

Ratio of Area of ∆ is Equal to (Base)².

→ Length = 4x = Base of ∆₁

→ Breadth = 3x = Base of ∆₂

→ Diagonal = 5x = Base of ∆₃ .

Hence,

→ Ratio of Area of ∆₁ : ∆₂ : ∆₃ = (4x)² : (3x)² : (5x)²

→ ∆₁ : ∆₂ : ∆₃ = 16x² : 9x² : 25x²

→ ∆₁ : ∆₂ : ∆₃ = 16 : 9 : 25 (Ans.)

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Answer 2

• Length = 4x

• Breadth = 3x

We know that,

Diagonal = √(l² + b²)

= √(4x² + 3x²)

= √(16x² + 9x²)

= √(25x²)

= 5x²

when,

• length =4x

★ Area of equilateral triangle = (√3/4) * (4x)²

When,

• Breadth = 3x

★ Area of equilateral triangle = (√3/4)* (3x)²

when,

• Diagonal = 5x

★ Area of equilateral triangle = (√3/4) * (5x)²

so ,

Ratio of Area of all 3 ∆'s = area of equaliteral ∆ = (√3/4) (3x)² : (√3/4)(4x)² : (√3/4)(5x)²

=> (3x)² : (4x)² : (5x)²

=> 9x² : 16x² : 25x²

=> 9 : 16 : 25