Question 15 of 42intermediate🔧 ApplyNumerical3 marks

One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area 128 cm², find the length of the shorter diagonal.

Correct Answer

Let the length of the shorter diagonal be x cm. Since one diagonal is twice as long as the other, the longer diagonal is 2x cm.

We know that the area of a rhombus is half the product of its diagonals. This can be derived by dividing the rhombus into four right-angled triangles using its diagonals. Therefore, Area = 1/2 × d₁ × d₂.

Substituting the given area of 128 cm², we get: 128 = 1/2 × x × 2x 128 = x² x = 128\sqrt{128} = 82\sqrt{2} cm.

Thus, the length of the shorter diagonal is 82\sqrt{2} cm.

Exercise: Exercise Set 6.2 | Q: 5 | (Page 25)
For More Understanding

Explanation

The question requires finding the diagonal of a rhombus given its area and the ratio of its diagonals. Although the provided context does not explicitly state the formula for the area of a rhombus, it discusses dividing shapes like trapeziums into triangles to find their areas. Similarly, a rhombus can be divided into four right-angled triangles by its diagonals, leading to the formula Area = 1/2 × d₁ × d₂. Using this formula and the given area of 128 cm², we set up the equation 1/2 × x × 2x = 128, which simplifies to x² = 128, yielding x = 82\sqrt{2} cm.

Solution Steps

  1. Step 1: Let the shorter diagonal be x cm. Then the longer diagonal is 2x cm.

  2. Step 2: Use the area formula for a rhombus: Area = 1/2 × d₁ × d₂.

  3. Step 3: Substitute the values into the formula: 128 = 1/2 × x × 2x.

  4. Step 4: Simplify the equation: 128 = x².

  5. Step 5: Solve for x: x = 128\sqrt{128} = 82\sqrt{2} cm.

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