Area - Study Mode

[#101] If a square of area $$frac{ ext{A}}{2}$$ is cut off from a given square of area A, then the ratio of diagonal of the cut off square to that of the given square is :
Correct Answer

(D) 1 : $$sqrt 2 $$

Explanation

Solution: Let the length of diagonal of the bigger square be x and that of the smaller square be y. Then, $$A = frac{1}{2}{x^2},,or,,x = sqrt {2A} $$ And, $$frac{A}{2} = frac{1}{2}{y^2},,or,,y = sqrt A $$ $$eqalign{
& herefore { ext{ Required ratio :}} cr
& = frac{y}{x} = frac{{sqrt A }}{{sqrt {2A} }} = 1:sqrt 2 { ext{ }} cr} $$

[#102] The length of a room is double its breadth. The cost of colouring the ceiling at Rs. 25 per sq. meter is Rs. 5000 and the cost of painting the four walls at Rs. 240 per sq. metre is Rs. 64800. Find the height of the room :
Correct Answer

(C) 4.5 m

Explanation

Solution: Let the breadth and height of the room be b metres and h metres respectively. Then, length of the room = (2b) metres Area of the ceiling : $$eqalign{
& = left( {2b imes b}
ight)m cr
& = left( {2{b^2}}
ight){m^2} cr} $$ $$eqalign{
& 2{b^2} = frac{{5000}}{{25}} cr
& 2{b^2} = 200 cr
& {b^2} = 100 cr
& b = 10 cr} $$ So, length = 20 m, breadth = 10 m Area of 4 walls : $$eqalign{
& = left[ {2left( {20 + 10}
ight) imes h}
ight]{m^2} cr
& = left( {60h}
ight){m^2} cr} $$ $$eqalign{
& herefore 60h = frac{{64800}}{{240}} cr
& Rightarrow 60h = 270 cr
& Rightarrow h = frac{{270}}{{60}} cr
& Rightarrow h = 4.5,m cr} $$

[#103] The sides of a triangle are in ratio of $$frac{1}{2}:frac{1}{3}:frac{1}{4}$$ xa0. If the perimeter is 52 cm, then the length of the smallest side is :
Correct Answer

(D) 12 cm

Explanation

Solution: Ratio of sides = $$frac{1}{2}:frac{1}{3}:frac{1}{4}$$ xa0 = 6 : 4 : 3 Perimeter = 52 cm So, sides are : $$eqalign{
& left( {52 imes frac{6}{{13}}}
ight)cm cr
& left( {52 imes frac{4}{{13}}}
ight)cm,& cr
& left( {52 imes frac{3}{{13}}}
ight)cm cr} $$ So, a = 24 cm, b = 16 cm, and c = 12 cm ∴ Length of smallest side = 12 cm

[#104] If x is the length of a median of an equilateral triangle, then its area is :
Correct Answer

(D) $$frac{{sqrt 3 }}{3}{x^2}$$

Explanation

Solution: Let the side of the triangle be a Then, $$eqalign{
& {a^2} = {left( {frac{a}{2}}
ight)^2} + {x^2} cr
& Leftrightarrow frac{{3{a^2}}}{4} = {x^2} cr
& Leftrightarrow {a^2} = frac{{4{x^2}}}{3} cr} $$ ∴ Area : $$eqalign{
& = frac{{sqrt 3 }}{4}{a^2} cr
& = frac{{sqrt 3 }}{4} imes frac{{4{x^2}}}{3} cr
& = frac{{{x^2}}}{{sqrt 3 }} cr
& = frac{{sqrt 3 {x^2}}}{3} cr} $$

[#105] If a square and a rhombus stand on the same base, then the ratio of the areas of the square and the rhombus is :
Correct Answer

(B) Equal to 1

Explanation

Solution: A square and a rhombus on the same base are equal in area.