Perimeter & Area of Plane Figures — Assertion-Reason Type Questions

Assertion (A): Length of the diagonal of a square field is $15\sqrt{2}$ m. Cost of ploughing the field at ₹20 per m² is ₹4,500.

Reason (R): $\text{Area of a Square} = \dfrac{(\text{Diagonal})^2}{4}$

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Diagonal length = $15\sqrt{2}$ m

For square :

Side = $\dfrac{\text{diagonal}}{\sqrt{2}}$

⇒ Side = $\dfrac{15\sqrt{2}}{\sqrt{2}}$

⇒ Side = 15 m.

Area = (side)²

= 15² = 225 m².

Cost of ploughing = ₹20 per m²

Total cost = Area × Cost per m²

= 225 × 20 = ₹4,500.

∴ Assertion (A) is true.

By formula,

$\text{Area of a Square} = \dfrac{(\text{Diagonal})^2}{2}$

∴ Reason (R) is false.

Assertion (A) is true, reason (R) is false.

Hence, option 1 is the correct option.

Assertion (A): An isosceles right triangle has area 8 cm². The length of its hypotenuse is 4 cm.

Reason (R): Area of an isosceles triangle with sides a, a and b is given by $\dfrac{1}{4}b\sqrt{4a^2 + b^2}$.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Let base and height of isosceles right triangle be a cm.

Area of triangle = $\dfrac{1}{2} \times \text{ base} \times \text{ height}$

⇒ 8 = $\dfrac{1}{2} \times a \times a$

⇒ 8 = $\dfrac{1}{2} \times a^2$

⇒ a² = 16

⇒ a = 4 cm.

By pythagoras theorem,

Hypotenuse² = Base² + Height²

⇒ Hypotenuse² = 4² + 4²

⇒ Hypotenuse² = 16 + 16

⇒ Hypotenuse² = 32

⇒ Hypotenuse = $\sqrt{32}$

⇒ Hypotenuse = $4\sqrt{2}$ cm.

∴ Assertion (A) is false.

Area of an isosceles triangle with sides a, a and b is given by $\dfrac{1}{4}b\sqrt{4a^2 - b^2}$.

∴ Reason (R) is false.

Hence, option 4 is the correct option.

Assertion (A): The area of rhombus is 84 cm². If its one diagonal is 7 cm, then side of the rhombus is 12 cm.

Reason (R): Area of rhombus is given by $\dfrac{1}{2}$ × product of its diagonals.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

By formula,

$\Rightarrow \text{Area of rhombus} = \dfrac{1}{2} \times d_1 \times d_2 \\[1em] \Rightarrow 84 = \dfrac{1}{2} × 7 × d_2 \\[1em] \Rightarrow d_2 = \dfrac{168}{7} \\[1em] \Rightarrow d_2 = 24 \text{ cm}.$

Now in rhombus diagonals bisect each other at right angles.

Thus,

By pythagras theorem,

(Side)² = $\Big(\dfrac{d_1}{2}\Big)^2 + \Big(\dfrac{d_2}{2}\Big)^2$

(Side)² = $\Big(\dfrac{7}{2}\Big)^2 + \Big(\dfrac{24}{2}\Big)^2$

(Side)² = (3.5)² + (12)²

(Side)² = 12.25 + 144

(Side)² = 156.25

Side = $\sqrt{156.25}$

Side = 12.5 cm

∴ Assertion (A) is false.

By formula,

Area of rhombus = $\dfrac{1}{2}$ × product of its diagonals

∴ Reason (R) is true.

Assertion (A) is false, reason (R) is true.

Hence, option 2 is the correct option.