Mensuration
- Perimeter of a rhombus is 2p unit and sum of length of diagonals is m unit, then area of the rhombus is
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Using Rule 12,
Side of a rhombus= 2p = p unit 4 2
OA = OC = y (let)
∴ AC = 2y units
OB = OD = x (let)
∴ BD = 2x units
From ∆OAB,
∠AOB = 90°
AB² = OA² + OB²⇒ p² = x² + y² 4
⇒ p² = 4x² + 4y² ...(i)
and 2x + 2y = m
On squaring both sides,
4x² + 4y² + 8xy = m²
⇒ p² + 8xy = m²
⇒ 8xy = m² – p²⇒ 4xy = 1 (m² + p²) 2 ∴ Area of the rhombus = 1 × AC × BD 2 = 1 × 2x × 2y = 1 × 4xy 2 2 = 1 × 1 (m² - p²) 2 2 = 1 (m² - p²) sq. units 4 Correct Option: C
Using Rule 12,
Side of a rhombus= 2p = p unit 4 2
OA = OC = y (let)
∴ AC = 2y units
OB = OD = x (let)
∴ BD = 2x units
From ∆OAB,
∠AOB = 90°
AB² = OA² + OB²⇒ p² = x² + y² 4
⇒ p² = 4x² + 4y² ...(i)
and 2x + 2y = m
On squaring both sides,
4x² + 4y² + 8xy = m²
⇒ p² + 8xy = m²
⇒ 8xy = m² – p²⇒ 4xy = 1 (m² + p²) 2 ∴ Area of the rhombus = 1 × AC × BD 2 = 1 × 2x × 2y = 1 × 4xy 2 2 = 1 × 1 (m² - p²) 2 2 = 1 (m² - p²) sq. units 4
