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41. The cost of painting the walls of a room 7
5 / 6
m long, 4 m wide at Rs 16.20 per sq m is Rs 1940.40. How high is the room?
  A.  3
7 / 5
m
  B.  4
2 / 3
m
  C.  5
2 / 3
m
  D.  3
4 / 5
m
     
   
View Answer

Shortcut:
(i) To find the area of the four walls of a room, if its length, breadth and height are given.
Area of the four walls of a room = 2 x (Length + Breadth) x Height.
(ii) To find the height of a room, if area of four walls of the room and its length and breadth are given.
Height =
Area of four walls of the room / 2(Length + Breadth)
metres
Here, Area of 4 floor =
1940.40 / 16.2
=
1078 / 9
sq m.
2h x
(
47 / 6
+ 5
)
=
1078 / 9

2h x
(
47 + 30 / 6
)
=
1078 / 9

h x
(
77 / 3
)
=
1078 / 9

h =
1078 x 3 / 77 x 9
=
14 / 3
= 4
2 / 3

Hence, the height is 4
2 / 3
metre.

42. Two square rooms, one a metre longer each way than the other, are of equal height, and cost respectively Rs 33600 and Rs 35280 to paper the walls at Rs 70 per sq m. Find the height.
  A.  5 m
  B.  3 m
  C.  6 m
  D.  8 m
     
   
View Answer

Shortcut:
(i) To find the area of the four walls of a room, if its length, breadth and height are given.
Area of the four walls of a room = 2 x (Length + Breadth) x Height.
(ii) To find the height of a room, if area of four walls of the room and its length and breadth are given.
Height =
Area of four walls of the room / 2(Length + Breadth)
metres
Here, Let the side of one square be y metre and the other by (y + 1) metre.
Now, as per question,
2h(y + y) =
33600 / 70
= 480
∴ hy = 120 ......(i)
2( y +1) x 2h =
35280 / 70
= 504
∴ yh + h = 126 ..........(ii)
Putting the value of yh from eq (i) into the equ (ii)
∴ h = 126 − 120 = 6m

43. One side of a parallelogram is 17 cm. The perpendiculr distance between this and the opposite side is 13 cm. Find the area of the parallelogram.
  A.  225 sq cm
  B.  210 sq cm
  C.  215 sq cm
  D.  221 sq cm
     
   
View Answer

Shortcut:
To find the area of a parallelogram if its Base and height are given.

Area of a parallelogram = Base x Height.
Here, base = 17, height = 13
Using these values in the shortcut, we get:
Area of parallelogram = 17 x 13 = 221
∴ Area of parallelogram is 221 sq cm

44. Find the area of a parallelogram whose two adjacent sides are 130 m and 140 m and one of the diagonals is 150 m long. Find also the cost of gravelling it at the rate of Rs 10 per sq metre.
  A.  16800 sq m; Rs 168000
  B.  15400 sq m; Rs 154000
  C.  18200 sq m; Rs 182000
  D.  15300 sq m; Rs 153000
     
   
View Answer

Shortcut:
To find the area of a parallelogram, if the lengths of the two adjacent sides and the length of the diagonal connecting the ends of the two sides are given.
where, a, b are the two adjacent sides and D is diagonal connecting the ends of the two sides.

Area of a parallelogram = 2√[s(s − a)(s − b)(s − D)]
where, s =
a + b + D / 2

Here, a = 130, b = 140, D = 150
So, s =
130 + 140 + 150 / 2
= 210
Using these values in the shortcut, we get:
Area of a parallelogram = 2√[210(210 − 130)(210 − 140)(210 − 150)]
= 2√(210 x 80 x 70 x 60)
= 2√(3 x 70 x 20 x 2 x 2 x 70 x 3 x 20)
= 2 x 3 x 70 x 20 x 2 = 16800
Cost of gravelling = Rs 10 per sq m
∴ Total cost = 10 x 16800 = 168000
∴ Area of parallelogram is 16800 sq m and cost of gravelling is Rs 168000

45. In a parallelogram the lengths of adjacent sides are 12 cm and 14 cm respectively. If the length of one diagonal is 16 cm, find the length of the other diagonal.
  A.  22.4 cm
  B.  15.4 cm
  C.  20.6 cm
  D.  24.8 cm
     
   
View Answer

Shortcut:
In a parallelogram, the sum of the squares of the diagonals = 2 x (the sum of the squares of the two adjacent sides)
or, D12 + D22 = 2(a2 + b2)
where, D1 and D2 are the diagonals and a & b are the adjacent sides.
Here, a = 12, b = 14, D1 = 16, D2 = ?
Using these values in the shortcut, we get:
(16)2 + D22 = 2[(12)2 + (14)2]
256 +D22 = 2(144 + 196)
D22 = 680 − 256
D22 = 424
D2 = √424 = 20.6
Hence, the length of other diagonal is 20.6 cm.

46. A parallelogram has an area of 160 sq cm. If the distance between its opposite sides are 10 cm and 16 cm. Find the sides of the parallelogram.
  A.  Length: 10 cm; Breadth: 14 cm
  B.  Length: 16 cm; Breadth: 10 cm
  C.  Length: 10 cm; Breadth: 15 cm
  D.  Length: 14 cm; Breadth: 18 cm
     
   
View Answer

Shortcut:
To find the sides of a parallelogram if the distance between its opposite sides and the area of the parallelogram is given.
Here, ABCD is a parallelogram, h1 and h2 are the distance between opposite sides, 'l' and 'b' are the sides of the parallelogram. 'A' is area of the parallelogram.

A = lh1 = bh2
∴ l =
A / h1
and b =
A / h2

Here, A = 160, h1 = 10, h2 = 16
Using these values in the shortcut, we get:
Length of parallelogram =
160 / 10
= 16
Breadth of parallelogram =
160 / 16
= 10
Hence, the length and breadth of parallelogram is 16 cm and 10 cm respectively.

47. In a rhombus, the length of the two diagonals are 40 m and 30 m respectively. Find its perimeter.
  A.  90 m
  B.  105 m
  C.  95 m
  D.  100 m
     
   
View Answer

Shortcut:
To find perimeter of a rhombus if the length of the two diagonals are given.
Perimeter of the rhombus = [2√(d12+d22)] units
where, d1 and d2 are the two diagonals.
Here, d1 = 40, d2 = 30
Using these values in the shortcut, we get:
Perimeter of the rhombus = 2√[(40)2+(30)2] units
= 2√(1600 + 900) = 2√2500 = 2 x 50 = 100
Hence, the perimeter of rhombus is 100 m.

48. The side and the height of a rhombus are 14 cm and 30 cm respectively. Find its area.
  A.  410 sq cm
  B.  400 sq cm
  C.  420 sq cm
  D.  450 sq cm
     
   
View Answer

Shortcut:
To find the area of a rhombus if the side and the height are givwen.
Area of rhombus = (side x height) sq units.
Here, side = 14, height = 30
Using these values in the shortcut, we get:
Area of rhombus = 14 x 30 = 420
Hence, the area of rhombus is 420 sq cm.

49. A rhombus of area 24 sq cm has one of its diagonals of 6 cm. Find the other diagonal and side of the rhombus.
  A.  Diagonal: 8cm; side: 5cm
  B.  Diagonal: 5cm; side: 3cm
  C.  Diagonal: 7cm; side: 10cm
  D.  Diagonal: 4cm; side: 6cm
     
   
View Answer

Shortcut:
To find the side and one of the diagonals of a rhombus if area and one of its diagonals are given
(i) Diagonal of the rhombus (d2) =
2A / d1

(ii) Side of the rhombus =
1 / 2
√[
d12 +
4A2 / d12
]

where A = Area of rhombus
d1 = length of the one diagonal
d2 = length of the other diagonal.

Here, A = 24, d1 = 6
Using these values in the shortcut, we get:
Diagonal of the rhombus (d2) =
2 x 24 / 6
= 8
Side of the rhombus =
1 / 2
√[
(6)2 +
4 x 24 x 24 / 6 x 6
]


=
1 / 2
√(36 + 4 x 4 x 4) =
1 / 2
√100 =
1 / 2
x 10 = 5
Hence, the other diagonal and side of the rhombus is 8 cm and 5 cm respectively.

50. One of the diagonals of a rhombus of side 5 cm measures 8 cm. Find the area of the rhombus.
  A.  22 sq cm
  B.  24 sq cm
  C.  28 sq cm
  D.  20 sq cm
     
   
View Answer

Shortcut:
If one of the diagonals of a rhombus of side 'y' units measures 'd' units, then
The area of the rhombus = d x
√[
y2 −
(
d / 2
)
2
]
sq units.
Length of the other diagonal = 2 x
√[
y2 −
(
d / 2
)
2
]
units.
[Note: if perimeter and one of the diagonals of a rhombus are given then,
y =
perimeter / 4


Here, y = 5, d = 8
Using these values in the shortcut, we get:
The area of the rhombus = 8 x
√[
52 −
(
8 / 2
)
2
]

= 8 x √(52 − 42)
= 8 x √(25 − 16)
= 8 x √(9) = 8 x 3 = 24
Hence, the area of the rhombus is 24 sq cm.

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