Perimeter and Area of Plane Figures

Q1: Fill in the blanks:

i. 100 m² = _______ cm²

Step 1: \(1 \, m = 100 \, cm\) \[ 1 \, m^2 = (100 \, cm)^2 = 10000 \, cm^2 \\ 100 \, m^2 = 100 \times 10000 = 1,000,000 \, cm^2 \] Answer: 1,000,000 cm²

ii. 5000 cm² = ________ dm²

Step 1: \(1 \, dm = 10 \, cm\) \[ 1 \, dm^2 = (10 \, cm)^2 = 100 \, cm^2 \\ 5000 \, cm^2 = \frac{5000}{100} = 50 \, dm^2 \] Answer: 50 dm²

iii. 1 hectare = ________ m²

Step 1: By definition: \[ 1 \, \text{hectare} = 10,000 \, m^2 \] Answer: 10,000 m²

iv. The area of a rhombus is half the product of its _________

Answer: diagonals

v. If the perimeter and area of a square are numerically equal then, the length of its side is ________.

Let the side of the square be \(s\). \[ \text{Perimeter} = 4s, \quad \text{Area} = s^2 \] Given: \[ 4s = s^2 \Rightarrow s^2 – 4s = 0 \Rightarrow s(s – 4) = 0 \\ s = 0 \text{ or } s = 4 \] Side cannot be zero.
Answer: 4

vi. The ratio of the lengths and breadth of a rectangle is 3 : 2 and its area is 96 cm². Then its length is ________.

Let length \(= 3x\), breadth \(= 2x\). \[ \text{Area} = l \times b = 3x \times 2x = 6x^2 = 96 \\ x^2 = \frac{96}{6} = 16 \Rightarrow x = 4 \\ \text{Length} = 3x = 3 \times 4 = 12 \, cm \] Answer: 12 cm