Area - Study Mode

[#231] The area of a square is three-fifths the area of a rectangle. The length of the rectangle is 25 cm and its breadth is 10 cm less than its length. What is the perimeter of the square ?
Correct Answer

(B) 60 cm

Explanation

Solution: Length of rectangle = 25 cm Breadth of rectangle = 15 cm Area of rectangle : $$eqalign{
& = left( {25 imes 15}
ight)c{m^2} cr
& = 375,cm cr} $$ Area of square : $$eqalign{
& = left( {frac{3}{5} imes 375}
ight)c{m^2} cr
& = 225,c{m^2} cr} $$ Side of square : $$eqalign{
& = sqrt {225} ,cm cr
& = 15,cm cr} $$ Perimeter of square : $$eqalign{
& = left( {4 imes 15}
ight)cm cr
& = 60,cm cr} $$

[#232] The area of a rectangle is thrice that of a square. If the length of the rectangle is 40 cm and its breadth is $$frac{3}{2}$$ times that of the side pf the square, then the side if the square is :
Correct Answer

(B) 20 cm

Explanation

Solution: Let the side of the square be x cm Then, its area = x 2 cm 2 Area of the rectangle = 3x 2 cm 2 ∴ 40 × $$frac{3}{2}$$ × x = 3x 2 ⇔ x = 20 cm

[#233] A rectangle becomes a square when its length is reduced by 10 units and its breadth is increased by 5 units. but by this process the area of the rectangle is reduced by 210 sq.units. The area of the rectangle (A) in square units is :
Correct Answer

(D) 2925 > A > 2900

Explanation

Solution: Let the length and breadth of the rectangle be $$l$$ and b respectively $$eqalign{
& l - 10 = b + 5 cr
& Rightarrow l - b = 15.....(i) cr
& { ext{And,}} cr
& Rightarrow lb - left( {l - 10}
ight)left( {b + 5}
ight) = 210 cr
& Rightarrow lb - left( {lb + 5l - 10b - 50}
ight) = 210 cr
& Rightarrow - 5l + 10b = 160 cr
& Rightarrow - l + 2b = 32.....(ii) cr} $$ Adding (i) and (ii), we get : b = 47 Putting b = 47 in equation (i), we get $$l$$ = 62 Hence, area of the rectangle : = $$lb$$ = (62 × 47) sq.units = 2914 sq.units Clearly, 2925 > A > 2900

[#234] The ratio of the areas of a square of side 6 cm and an equilateral triangle of side 6 cm is :
Correct Answer

(D) 4 : $$sqrt 3 $$

Explanation

Solution: Required ratio : $$eqalign{
& = left( {6 imes 6}
ight):left( {frac{{sqrt 3 }}{4} imes 6 imes 6}
ight) cr
& = 4:sqrt 3 cr} $$

[#235] One side of a right-angled triangle is twice the other, and the hypotenuse is 10 cm. The area of the triangle is :
Correct Answer

(A) 20 cm 2

Explanation

Solution: Let the sides be a cm and 2a cm Then, $$eqalign{
& {a^2} + {left( {2a}
ight)^2} = {left( {10}
ight)^2} cr
& Rightarrow 5{a^2} = 100 cr
& Rightarrow {a^2} = 20 cr} $$ ∴ Area : $$eqalign{
& = left( {frac{1}{2} imes a imes 2a}
ight) cr
& = {a^2} cr
& = 20,c{m^2} cr} $$