Area of a square is 8 times the area of a rectangle. Find the ratio of diagonal of the square to the diagonal of the rectangle, if the length of the rectangle is half the length of the side of the square.
1. 1 : 2
2. 3√2 : 5√3
3. 4√2 : √5
4. √3 : √5
Area of a square is 8 times the area of a rectangle. Find the ratio of diagonal of the square to the diagonal of the rectangle, if the length of the rectangle is half the length of the side of the square.
1. 1 : 2
2. 3√2 : 5√3
3. 4√2 : √5
4. √3 : √5
1. 1 : 2
2. 3√2 : 5√3
3. 4√2 : √5
4. √3 : √5
Correct Answer – Option 3 : 4√2 : √5
Given :
Area of the square is 8 times the area of the rectangle
Length of rectangle is half the length of the side of the square
Formula Used :
Area of the square = side2
Area of the rectangle = length × breadth
Calculations :
Let the side of the square be ‘s’
Let the length of the rectangle be ‘l’ and breadth be ‘b’
According to the question
S2 = 8 × (I × b)
(2l)2 = 8 × (l × b) (l = (s/2) or s = 2 l)
4 l2 = 8 × l × b
⇒ b = l/2
Now,
Diagonal of the square = √2 × side = √2s = 2√2 l (s = 2l)
Diagonal of the rectangle = √[(l)2 + (b)2] = √[(l)2 + (l/2)2]
⇒ √[(5/4)(l)2]
⇒ (l/2)√5
Diagonal of square : Diagonal of rectangle = 2√2l : (l/2)√5
⇒ 4√2 : √5
∴ The ratio of the diagonals is 4√2 : √5