Area of a general quadrilateral | Mensuration | Assessment | English | Grade 8

If the length of a diagonal of a quadrilateral is d and the perpendiculars dropped on it from the remaining opposite vertices are denoted by h₁ and h₂ respectively, then the area of the quadrilateral is _______________.

• ½ × d × (h₁ - h₂)

• ½ × d × (h₁/h₂)

• ½ × d × (h₁ + h₂)

• ½ × d × (h₁ × h₂)

The diagonal of a quadrilateral 14 cm long and length of the perpendiculars dropped on it from the remaining opposite vertices are 9 cm and 11 cm. The area of the quadrilateral is ________.

• 150 cm²

• 140 cm²

• 145 cm²

• 155 cm²

Answer explanation

Area of quadrilateral = ½ × d × (h₁ + h₂) = ½ × 14 × (9 + 11) = 140 cm² So, Option 2 is correct.

If the area of a quadrilateral is 438 cm² and the length of perpendiculars dropped on a diagonal from the remaining opposite vertices are 11 cm and 13 cm respectively, then what is the length of the diagonal?

• 37.5 cm

• 39 cm

• 42 cm

• 36.5 cm

Answer explanation

Area of quadrilateral = ½ × d × (h₁ + h₂) According to the question, we have, 438 = ½ × d × (11 + 13) 438 = 12 d So, d = 438/ 12 = 36.5 cm Hence, Option 4 is correct.

What is the area of the adjoining quadrilateral?

• 96 cm²

• 112.5 cm²

• 120 cm²

• 127.5 cm²

Answer explanation

Length of diagonal of quadrilateral can be found by pythagoras theorem hence 8² + d² = 17² d² = 17² - 8² = 225 so, d = 15 cm Area of quadrilateral = ½ × d × (h₁ + h₂) = ½ × 15 × (8 + 7) = 112.5 cm² So, option 2 is correct.

If the diagonal d of a quadrilateral is doubled and the heights h₁ and h₂ falling on d are halved, then the area of quadrilateral is __________.

• Half of the area of the original quadrilateral

• One-fourth the area of the original quadrilateral

• Equal to the area of the original quadrilateral

• Twice the area of the original quadrilateral

Answer explanation

Let the diagonal of the quadrilateral be d and the heights falling on d be h₁ and h₂, respectively. So, area of the original quadrilateral = ½ × d × (h₁ + h₂) ......(1) Now assume that the length of the diagonal is doubled, i.e., d' = 2d and the heights are halved, i.e., h'₁ = h₁/2 and h'₂ = h₂/2 So, area of the new quadrilateral = ½ × 2d × (h'₁ + h'₂) = d × (h₁/2 + h₂/2) = ½ × d × (h₁ + h₂) ......(2) (On taking 1/2 as common) From (1) and (2), we can see that the area of the quadrilateral remains the same. Hence, option 3 is correct.

Raju has a farm of the shape shown in the adjoining figure. He wants to plough his farm and also fence it. If the cost of fencing is ₹ 150 per metre and the cost of ploughing is ₹ 250 per square metre, then how much money does Raju need?

• ₹ 9000

• ₹ 66500

• ₹ 75500

• ₹ 79700

Answer explanation

For fencing we need to find the perimeter of the farm Perimeter of the farm = Circumference of upper semicircle + Circumference of lower semicircle + 2 (breadth of rectangle) = (1/2) × 2 × (22/7) × 7 + (1/2) × 2 × (22/7) × 7 + 2(8) = 22 + 22 + 16 = 60 m Cost of fencing = ₹ 60 × 150 = ₹ 9000 For finding the cost of ploughing we have to find area of farm. so area of farm = area of rectangle + 2 ×(1/2) area of circle = (14 × 8) + [2 ×(1/2) × (22/7) × 7 × 7] = 112 + 22 × 7 = 112 + 154 = 266 m² Cost of ploughing = 266 × 250 = ₹ 66500 Total money Raju needs = Cost of fencing + Cost of ploughing = ₹ 9000 + ₹ 66500 = ₹ 75500 Hence option 3 is correct answer.