Perimeter & Area of Plane Figures — Multiple Choice Questions

The area of an equilateral triangle is $16\sqrt{3}$ cm². The length of its each side is :

1. 4 cm

2. 6 cm

3. 8 cm

4. 9 cm

Answer

Let each side of equilateral triangle be a cm.

Area of equilateral triangle = $\dfrac{\sqrt{3}}{4}$ a²

⇒ $16\sqrt{3} = \dfrac{\sqrt{3}}{4}$ a²

⇒ 64 = a²

⇒ a = $\sqrt{64}$

⇒ a = 8 cm.

Hence, option 3 is the correct option.

The length of the base and the area of an isosceles triangle are respectively 8 cm and 12 cm². The length of each equal side is :

1. 15 cm

2. 10 cm

3. 6 cm

4. 5 cm

Answer

ABC is an isosceles triangle with AB = AC and base (BC) = 8 cm.

Area of a triangle = $\dfrac{1}{2}$ × base × height

Area of triangle ABC = $\dfrac{1}{2}$ × BC × AD

⇒ 12 = $\dfrac{1}{2}$ × 8 × AD

⇒ AD = $\dfrac{24}{8}$ = 3 cm.

In an isosceles triangle, the perpendicular from the common vertex to the base, bisects the base.

∴ BD = DC = $\dfrac{BC}{2} = \dfrac{8}{2}$ = 4 cm.

In triangle ABD,

⇒ Hypotenuse² = Base² + Height²

⇒ AB² = BD² + AD²

⇒ AB² = 4² + 3²

⇒ AB² = 16 + 9

⇒ AB² = 25

⇒ AB = $\sqrt{25}$

⇒ AB = 5 cm.

∴ AB = AC = 5 cm.

Hence, option 4 is the correct option.

The length of the base of an isosceles triangle is 24 cm and the length of each equal side is 13 cm. The area of the triangle is :

1. 60 cm²

2. 120 cm²

3. 156 cm²

4. 164 cm²

Answer

Given,

Length of each equal side (a) = 13 cm and base (b) = 24 cm.

By formula,

$\Rightarrow \text{Area of isosceles triangle} = \dfrac{1}{4}b \sqrt{4a^2 - b^2} \\[1em] = \dfrac{24}{4} \sqrt{4(13)^2 - 24^2} \\[1em] = 6 \sqrt{4 × 169 - 576} \\[1em] = 6 \sqrt{676 - 576} \\[1em] = 6 \sqrt{100} \\[1em] = 6 × 10 \\[1em] = 60 \text{ cm}^2.$

Hence, option 1 is the correct option.

One side of a parallelogram is 6 m. If the area of a parallelogram is 30 m², then its height corresponding to the side is :

1. 10 m

2. 5 m

3. 12 m

4. 14 m

Answer

Area of //gm = Base × Height

⇒ 30 = 6 × Height

⇒ Height = $\dfrac{30}{6}$ = 5 m.

Hence, option 2 is the correct option.

The length of a diagonal of a square is 8 cm. The area of the square is :

1. 64 cm²

2. 32 cm²

3. 128 cm²

4. 256 cm²

Answer

Area of square = $\dfrac{1}{2}$ × (diagonals)²

= $\dfrac{1}{2}$ × (8)²

= $\dfrac{1}{2}$ × 64

= 32 cm².

Hence, option 2 is the correct option.

The length of one diagonal of a rhombus of area 24 cm² is 6 cm. The length of the other diagonal is :

1. 4 cm

2. 8 cm

3. 12 cm

4. 16 cm

Answer

By formula,

Area of rhombus = $\dfrac{1}{2}$ × d₁ × d₂

⇒ 24 = $\dfrac{1}{2}$ × 6 × d₂

⇒ d₂ = $\dfrac{48}{6}$ = 8 cm.

Hence, option 2 is the correct option.

The distance between the parallel sides of a trapezium of area 27.5 cm² is 5 cm. If the length of one of those two parallel sides is 7.5 cm, then the length of the other side is :

1. 3.5 cm

2. 7 cm

3. 8.5 cm

4. 9 cm

Answer

Let the length of other parallel side be a cm.

By formula,

Area of trapezium = $\dfrac{1}{2}$ × (sum of parallel sides) × distance between parallel sides

⇒ 27.5 = $\dfrac{1}{2}$ × (7.5 + a) × 5

⇒ 55 = (7.5 + a) × 5

⇒ 7.5 + a = $\dfrac{55}{5}$

⇒ 7.5 + a = 11

⇒ a = 11 - 7.5 = 3.5 cm.

Hence, option 1 is the correct option.

The area of the triangle whose three sides are 13 cm, 14 cm and 15 cm, is :

1. 72 cm²

2. 84 cm²

3. 92 cm²

4. 96 cm²

Answer

Let a = 13 cm, b = 14 cm and c = 15 cm.

s = $\dfrac{a + b + c}{2}$

= $\dfrac{13 + 14 + 15}{2}$

= $\dfrac{42}{2}$ = 21.

By formula,

Area of triangle = $\sqrt{s(s - a)(s - b)(s - c)}$

$= \sqrt{21(21 - 13) \times (21 - 14) \times (21 - 15)} \\[1em] = \sqrt{21 \times (8) \times (7) \times (6)} \\[1em] = \sqrt{7056} \\[1em] = 84 \text{ cm}^2.$

Hence, option 2 is the correct option.

The area of the equilateral triangle of side 4 cm is :

1. 6.928 cm²

2. 6.298 cm²

3. 5.928 cm²

4. 5.528 cm²

Answer

Side (a) = 4 cm.

Area of an equilateral triangle = $\dfrac{\sqrt{3}}{4}$ × a²

= $\dfrac{\sqrt{3}}{4}$ × (4)²

= $\dfrac{\sqrt{3}}{4}$ × 16

= $4\sqrt{3}$

= 4 × 1.732 = 6.928 cm².

Hence, option 1 is the correct option.

If the height of an equilateral triangle is $12\sqrt{3}$ cm, then its area is :

1. $44\sqrt{3}$ cm²

2. $144\sqrt{3}$ cm²

3. $140\sqrt{3}$ cm²

4. $132\sqrt{3}$ cm²

Answer

Let the length of each side of the equilateral triangle be a cm.

We know that,

Height of an equilateral triangle = $\dfrac{\sqrt{3}}{2}$ side

⇒ $12\sqrt{3} = \dfrac{\sqrt{3}}{2}a$

⇒ a = 24 cm.

$\Rightarrow \text{Area of an equilateral triangle} = \dfrac{\sqrt{3}}{4} \times a^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times (24)^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times 24 \times 24 \\[1em] = \sqrt{3} \times 24 \times 6 \\[1em] = 144 \sqrt{3} \text{ cm}^2.$

Hence, option 2 is the correct option.

The lengths of two adjacent sides of a parallelogram are 12 cm and 10 cm respectively. If the distance between the longer sides be 8 cm, then the distance between two shorter sides will be :

1. 9.1 cm

2. 9.2 cm

3. 9.4 cm

4. 9.6 cm

Answer

Longer side = 12 cm

Shorter side = 10 cm

Distance between longer sides (height corresponding to 12 cm) = 8 cm

Area of //gm = Base × Height

= 12 × 8 = 96 cm².

Distance between the shorter side be h cm.

Area of //gm = Base × Height

⇒ 96 = 10 × h

⇒ h = $\dfrac{96}{10}$ = 9.6 cm.

Hence, option 4 is the correct option.

The ratio of heights of two triangles is 2 : 3 and the ratio of their areas 3 : 2. The ratio of their bases is :

1. 1 : 4

2. 2 : 8

3. 9 : 4

4. 4 : 9

Answer

Let b₁ and b₂ be the bases of the triangles.

Heights (h₁ : h₂) = 2 : 3

Areas (A₁ : A₂) = 3 : 2

Area of a triangle = $\dfrac{1}{2}$ × Base × Height

$\Rightarrow \dfrac{A_1}{A_2} = \dfrac{\dfrac{1}{2} \times b_1 × h_1}{\dfrac{1}{2} \times b_2 × h_2} \\[1em] \Rightarrow \dfrac{3}{2} = \dfrac{b_1}{b_2} \times \dfrac{h_1}{h_2} \\[1em] \Rightarrow \dfrac{3}{2} = \dfrac{b_1}{b_2} \times \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{b_1}{b_2} = \dfrac{3 × 3}{2 × 2} \\[1em] \Rightarrow \dfrac{b_1}{b_2} = \dfrac{9}{4} \\[1em] \Rightarrow b_1 : b_2 = 9 : 4.$

Hence, option 3 is the correct option.

The perimeter of a rectangle is 26 m. If the length of the rectangle is 3 m more than its breadth, then its area will be :

1. 36 m²

2. 38 m²

3. 40 m²

4. 42 m²

Answer

By formula,

Perimeter of a rectangle = 2(l + b)

⇒ 26 = 2(3 + b + b)

⇒ 26 = 2(3 + 2b)

⇒ 3 + 2b = $\dfrac{26}{2}$

⇒ 3 + 2b = 13

⇒ 2b = 13 - 3

⇒ 2b = 10

⇒ b = $\dfrac{10}{2}$ = 5 m.

⇒ l = 3 + b = 3 + 5 = 8 m.

Area of rectangle = l × b

= 8 × 5 = 40 m².

Hence, option 3 is the correct option.

If the length of two diagonals of a rhombus are 6 cm and 4 cm, then its area is :

1. 10 cm²

2. 12 cm²

3. 14 cm²

4. 16 cm²

Answer

Area of rhombus = $\dfrac{1}{2}$ × d₁ × d₂

= $\dfrac{1}{2}$ × 6 × 4

= 3 × 4 = 12 cm².

Hence, option 2 is the correct option.