A rectangle is also a parallelogram.
AO = CO
BO = DO
AC = BD
Problem 1 :
The diagonal of the TV screen is 82 cm, and the height is 40 cm. Calculate the width of the screen.
Solution :
The diagonal will divide the rectangle into two equal right triangles.
Width of the screen = length of the screen
(diagonal)^{2} = (width)^{2} + (length)^{2}
82^{2} = (width)^{2} + (40)^{2}
6724 = (width)^{2} + 1600
Subtracting 1600 on both sides, we get
5124 = (width)^{2}
Width = √5124
width = 71.58
Problem 2 :
The aspect ratio of the rectangle and its diagonal is 9 : 12 : 15. Calculate the area of the rectangle if the length of the diagonal is 105 cm.
Solution :
Length = 9x, width = 12x and diagonal = 15x
Length of the diagonal = 105 cm
15x = 105
x = 105/15
x = 7
Length = 9(7) ==> 63
width = 12(7) ==> 84
Area of rectangle = 63 (84)
= 5292 cm^{2}
Problem 3 :
There is a rectangle with the length of 12 cm and diagonal 8 cm longer than the width. Calculate the area of the rectangle.
Solution :
Let x be the width of the rectangle.
diagonal = 8 + x, length = 12 cm
(8+x)^{2} = x^{2} + 12^{2}
64 + 2x + x^{2} = x^{2} + 144
64 + 2x = 144
2x = 144 - 64
2x = 80
x = 40
Area of rectangle = 12x
= 12(40)
= 480 cm^{2}
Problem 4 :
The length of the sides of the rectangular garden are in the ratio 1 : 2. The connection of the centers of the adjacent sides is 20 m long. Calculate the perimeter of the rectangle.
Solution :
Length = x and width = 2x
Distance between centers = 20 m
x^{2} + (2x)^{2} = 20^{2}
x^{2} + 4x^{2} = 400
5x^{2} = 400
x^{2} = 80
x = 4√5
Perimeter of the rectangle = 2(x + 2x)
= 2(3x)
= 6(4√5)
= 24√5
Problem 5 :
The dimension of the rectangular plot are (x + 1) m and (2x - y)m. The sum of x and y is 3 m and the perimeter of the plots is 36 m. Find the area of the diagonal of the plot.
Solution :
Length = x +1, width = 2x - y
x + y = 3
y = 3 - x
Perimeter = 36
2(x + 1 + 2x - y) = 36
2(3x - y + 1) = 36
(3x - y + 1) = 18
3x - (3 - x) + 1 = 18
3x - 3 + x = 17
4x = 20
x = 5
Then y = 3 - 5 ==> -2(doesn't mean the measure)
==> 5 + 1 ==> 6 m
Length = x + 1 Applying x = 5 length = 5 + 1 = 6 |
width = 2x - y Applying x = 5 width = 2(5) - 2 = 10 - 2 width = 8 |
Area of the plot = 6(8)
Area of plot = 48 square meter.
Problem 6 :
The diagonal of the rectangle given below is 36 m. Find the values of x and y.
Solution :
Diagonals will bisect each other
So, 2x+4y = 4x-y
2x-4x + 4y + y = 0
-2x + 5y = 0 ---(1)
Length of diagonal = 36
2x + 4y + 4x - y = 36
6x + 3y = 36
Dividing by 3 on both sides.
2x + y = 12 ---(2)
(1) + (2)
6y = 12
y = 2
Applying y = 2 in (2), we get
2x + 2 = 12
2x = 10
x = 5
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM