The diagonals of a rhombus are in the ratio $3\ :\ 4$ . If its area is $150$ cm $^2$ , find the lengths of the diagonals.

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1. Appropriate Question :

   The diagonals of a rhombus are in the ratio 3 : 4. If its area is 150 cm² , find the lengths of the diagonals.

   Given :

   • Ratio of the diagonals of the rhombus = 3 : 4

   • Area of the diagonal = 150 cm²

   To find :

   • Length of its diagonals

   Concept :

   Here, we have to find the length of the diagonals of the rhombus. For that firstly we will assume the length of the diagonals according to the ratio mentioned in the question. Then by using the formula of the area of rhombus we will get our required answer.

   Formula to find area of rhombus :-

   • Area of rhombus = ½ × d₁ × d₂

   where,

   • d₁ denotes the length of the first diagonal

   • d₂ denotes the length of the second diagonal

   Solution :

   Let us assume the length of the diagonals be 3x and 4x.

   • Length of the first diagonal = 3x
   • Length of the second diagonal = 4x

   → Area of rhombus = ½ × d₁ × d₂

   → Substituting the given values :-

   → 150 = ½ × 3x × 4x

   → 150 × 2 = 3x × 4x

   → 300 = 12x²

   → 300 ÷ 12 = x²

   → 25 = x²

   → Taking square root on both the sides :-

   → √25 = x

   → √(5 × 5) = x

   → ± 5 = x

   → As we know, length of diagonal cannot be negative. The negative sign will get rejected.

   → ± 5 Reject – ve = x

   → 5 = x

   → The value of x = 5

   Substitute the value of x in the length of diagonals :-

   FIRST DIAGONAL :-

   → Length of the first diagonal = 3x

   → Length of the first diagonal = 3 × 5

   → Length of the first diagonal = 15 cm

   SECOND DIAGONAL :-

   → Length of the second diagonal = 4x

   → Length of the second diagonal = 4 × 5

   → Length of the second diagonal = 20 cm

   Therefore, the length of the diagonals of the rhombus are 15 cm and 20 cm.

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