(a) The diagonals of a rhombus are 24 cm and 10 cm. Calculate its area and perimeter.

(b) The diagonals of a field in the form of a quadrilateral are 106 m and 80 m and intersect each other at right angles. Find the cost of cultivating the field at the rate of ₹ 25.50 per 100 m².

(a)

Let ABCD be the rhombus with diagonals AC and BD measuring 24 cm and 10 cm, respectively. We know that the diagonals of a rhombus bisect each other at right angles.

Area of rhombus = $\dfrac{1}{2}$ x (Product of diagonals)

= $\dfrac{1}{2}$ x 24 x 10

= $\dfrac{1}{2}$ x 240

= 120 cm²

Applying the Pythagorean theorem in Δ ADE:

AD² = AE² + DE²

= 12² + 5²

= 144 + 25

= 169

AD = $\sqrt{169}$

= 13 cm

Perimeter of the rhombus = 4 x side

= 4 x 13

= 52 cm

Hence, the area of the rhombus is 120 cm² and the perimeter is 52 cm.

(b) When the diagonals of a quadrilateral intersect at right angles, the area can be found using the formula:

Area of the quadrilateral = $\dfrac{1}{2}$ x d₁ x d₂

= $\dfrac{1}{2}$ x 106 x 80

= $\dfrac{1}{2}$ x 8480

= 4240 m²

Given that the cost of cultivation is ₹ 25.50 per 100 m².

$\text{Total cost} = \dfrac{\text{Total area}}{100} \times \text{Rate per } 100 \text{ m}^2$

= $\dfrac{4240 \times 25.5}{100}$

= ₹ 1081.20

Hence, the total cost of cultivation is ₹ 1081.20.