A rectangle is such that its area is 48 cm² and its perimeter is 28 cm. If a rhombus is formed by joining the midpoints of the sides of this rectangle, what is the area of the rhombus?

Correct — B. Find the sides first. The perimeter gives l + b = 28 ÷ 2 = 14, and the area gives l × b = 48.

Sum 14, product 48: t² − 14t + 48 = 0
Factorise: (t − 6)(t − 8) = 0, so the sides are 8 cm and 6 cm

Each rhombus diagonal joins the midpoints of two opposite sides
So the diagonals equal the rectangle's sides: 8 cm and 6 cm
Rhombus area = ½ × 8 × 6 = 24 cm² → option (b)

Why the others are wrong

• (a)
  12 cm² — 12 cm² is a quarter of the rectangle. The four corner triangles are each ½ × 4 × 3 = 6 cm², so they remove 24 and leave 24, half of 48.
• (c)
  48 cm² — 48 cm² is the rectangle itself. The rhombus sits inside it with diagonals equal to the sides, so its area is ½ × 8 × 6 = 24.
• (d)
  20 cm² — 20 is the rhombus's perimeter in cm, not its area: each side is √(4² + 3²) = 5 cm, and 4 × 5 = 20.

A rectangle is fixed by the sum and product of its sides: half the perimeter gives l + b, the area gives l × b, and the sides are the roots of t² − (l + b)t + lb = 0.

Joining the midpoints of a rectangle's sides gives a rhombus whose diagonals are the rectangle's length and breadth. Rhombus area = ½ × d₁ × d₂ = ½ × l × b.

So the rhombus is always half the rectangle, 48 ⁄ 2 = 24 cm². The perimeter is not needed for the area; it fixes the rhombus's side, 5 cm.

Key facts

• Joining the midpoints of a rectangle's sides gives a rhombus with half the rectangle's area.
• Area of a rhombus = ½ × product of its diagonals.
• Here the rectangle is 8 cm × 6 cm and the rhombus has side 5 cm.

Common traps

• Stopping at the rectangle's area, 48, because the rhombus sits inside it.
• Taking the rhombus's side or perimeter for its area.

No directly related past PYQ was found.