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51. If the perimeter of a rhombus is 4a and lengths of the diagonals are x and y, then its area is:
  A.  
1 / 2
xy
  B.  
2 / 3
xy
  C.  
1 / 5
xy
  D.  
3 / 4
xy
     
   
View Answer

Shortcut:
To find the area of a rhombus if its diagonals are given.
The area of the rhombus =
1 / 2
x D1 x D2
where, D1 & D2 are its diagonals.

Here, D1 = x, D2 = y
Using these values in the shortcut, we get:
The area of the rhombus =
1 / 2
(x x y)
Hence, the area of the rhombus is
1 / 2
xy.

52. In a rhombus whose area is 144 sq cm one of its diagonals is twice as long as the other. The length of its diagonals are:
  A.  15 cm, 20 cm
  B.  12 cm, 24 cm
  C.  13 cm, 26 cm
  D.  14 cm, 28 cm
     
   
View Answer

Shortcut:
To find the area of a rhombus if its diagonals are given.
The area of the rhombus =
1 / 2
x D1 x D2
where, D1 & D2 are its diagonals.
Here, Area = 144, D1 = 2D2
Using these values in the shortcut, we get:
The area of the rhombus =
1 / 2
x 2D2 x D2
144 = D2 x D2
144 = (D2)2
or, D2 = √144 = 12
So, D1 = 2D2 = 2 x 12 = 24
Hence, the lengths of diagonals are 12 cm and 24 cm.

53. Find the side of a rhombus one of whose diagonals measure 6 cm and the other 8 cm.
  A.  9 cm
  B.  5 cm
  C.  10 cm
  D.  3 cm
     
   
View Answer

Shortcut:
To find the sides of the rhombus if its two diagonals are given.
Side of the rhombus =
1 / 2
x √[(D1)2 + (D2)2]
where, D1 & D2 are its diagonals.
Here, D1 = 6, D2 = 8
Using these values in the shortcut, we get:
Side of the rhombus =
1 / 2
x √[(6)2 + (8)2]
=
1 / 2
x √(36 + 64)
=
1 / 2
x √(100)
=
1 / 2
x 10 = 5
Hence, the side of a rhombus is 5 cm.

54. A trapezium has the perpendicular distance between the two parallel sides 60 m. If the lengths of the parallel sides be 40 m and 130 m, then find the area of trapezium.
  A.  5100 sq m
  B.  5500 sq m
  C.  5000 sq m
  D.  5350 sq m
     
   
View Answer

Shortcut:
To find the area of a trapezium, when length of parallel sides and the perpendicular distance between them is given.
Area of a trapezium =
1 / 2
x (sum of parallel sides x perpendicular distance between the parallel sides)
or Area of a trapezium =
1 / 2
x (a + b) x h
where, 'a' & 'b' are the parallel sides of a trapezium and 'h' is the perpendicular distance between 'a' & 'b'.
Here, a = 130, b = 40, h = 60
Using these values in the shortcut, we get:
Area of a trapezium =
1 / 2
x (130 + 40) x 60
=
1 / 2
x 170 x 60 = 170 x 30 = 5100
Hence, the side of the trapezium is 5100 m.

55. The area of a trapezium is 384 sq cm. If its parallel sides are in the raio 3 : 5 and the perpendicular distance between them be 12 cm, the smaller of parallel sides is:
  A.  14 cm
  B.  24 cm
  C.  34 cm
  D.  28 cm
     
   
View Answer

Shortcut:
To find the area of a trapezium, when length of parallel sides and the perpendicular distance between them is given.
Area of a trapezium =
1 / 2
x (sum of parallel sides x perpendicular distance between the parallel sides)
or Area of a trapezium =
1 / 2
x (a + b) x h
where, 'a' & 'b' are the parallel sides of a trapezium and 'h' is the perpendicular distance between 'a' & 'b'.
Here, a = 3y, b = 5y, h = 12, Area = 384
Using these values in the shortcut, we get:
Area of a trapezium =
1 / 2
x (3y + 5y) x 12
384 =
1 / 2
x 8y x 12
y =
384 x 2 / 8 x 12
= 8
∴ smaller side = 3y = 3 x 8 = 24
Hence, the smaller side of the trapezium is 24 cm.

56. The cross section of a canal is a trapezium in shape. If the canal is 10 m wide at the top and 6 m wide at the bottom and the area of cross section is 640 sq m, the length of the canal is:
  A.  80 m
  B.  60 m
  C.  70 m
  D.  90 m
     
   
View Answer

Shortcut:
To find the area of a trapezium, when length of parallel sides and the perpendicular distance between them is given.
Area of a trapezium =
1 / 2
x (sum of parallel sides x perpendicular distance between the parallel sides)
or Area of a trapezium =
1 / 2
x (a + b) x h
where, 'a' & 'b' are the parallel sides of a trapezium and 'h' is the perpendicular distance between 'a' & 'b'.
Here, Let the length of canal be y metre.
a = 10, b = 6, h = y
Using these values in the shortcut, we get:
Area of a trapezium =
1 / 2
x (10 + 6) x y
640 =
1 / 2
x 16y
640 = 8y
or, y =
640 / 8
= 80
Hence, the length of the canal is 80 m.

57. In a trapezium, parallel sides are 60 and 90 cm respectively and non-parallel sides are 40 and 50 cm respectively. Find its area.
  A.  2000 sq cm
  B.  2200 sq cm
  C.  2500 sq cm
  D.  3000 sq cm
     
   
View Answer

Shortcut:
To find the area of a trapezium, when the lengths of parallel sides and non-parallel sides are given.
Area of a trapezium =
a + b / k
x √[s(s − k)(s − c)(s − d)]
where, k = (a − b) i.e., the difference between the parallel sides and c and d are the two non-parallel sides of the trapezium.
And s =
k + c + d / 2

Here, a = 60, b = 90, c = 40, d = 50
k = 90 − 60 = 30
Then, s =
30 + 40 + 50 / 2
=
120 / 2
= 60
Using these values in the shortcut, we get:
Area of a trapezium =
60 + 90 / 30
x √[60(60 − 30)(60 − 40)(60 − 50)]
=
150 / 30
x √(60 x 30 x 20 x 10)
= 5 x √(2 x 30 x 30 x 20 x 10)
= 5 x 20 x 30 = 3000
Hence, the area of trapezium is 3000 sq cm.

58. A field is in the form of a trapezium whose parallel sides are 120 m and 75 m and the non-parallel sides are 105 m and 72 m. Find the cost of ploughing the field at the rate of 60 paise per sq metres.
  A.  Rs 3000
  B.  Rs 3210
  C.  Rs 3401
  D.  Rs 3500
     
   
View Answer

Shortcut:
To find the area of a trapezium, when the lengths of parallel sides and non-parallel sides are given.
Area of a trapezium =
a + b / k
x √[s(s − k)(s − c)(s − d)]
where, k = (a − b) i.e., the difference between the parallel sides and c and d are the two non-parallel sides of the trapezium.
And s =
k + c + d / 2

Here, a = 120, b = 75, c = 105, d = 72
k = 120 − 75 = 45
Then, s =
45 + 105 + 72 / 2
=
222 / 2
= 111
Using these values in the shortcut, we get:
Area of a trapezium =
120 + 75 / 45
x √[111(111 − 45)(111 − 105)(111 − 72)]
=
195 / 45
x √(111 x 66 x 6 x 39)
= 4.33 x √(111 x 66 x 6 x 39)
= 4.33 x 1309 = 5668
∴ cost of ploughing the field =
5668 x 60 / 100
= 3401
Hence, the cost is Rs 3401.

59. In a trapezium parallel sides are 60 and 90 cm respectively and non-parallel sides are 40 and 50 cm respectively. Find the perpendicular distance between the two parallel sides of the trapezium.
  A.  50 cm
  B.  40 cm
  C.  45 cm
  D.  30 cm
     
   
View Answer

Shortcut:
To find the perpendicular distance between the two parallel sides of the trapezium.
Perpendicular distance =
2 / k
x √[s(s − k)(s − c)(s − d)]
where, k = (a − b) i.e., the difference between the parallel sides and c and d are the two non-parallel sides of the trapezium.
And s =
k + c + d / 2

Here, a = 60, b = 90
k = 90 − 60 = 30
c = 40 & d = 50
Then, s =
30 + 40 + 50 / 2
=
120 / 2
= 60
Using these values in the shortcut, we get:
Perpendicular distance =
2 / 30
x √[60(60 − 30)(60 − 40)(60 − 50)]
=
1 / 15
x √(60 x 30 x 20 x 10)
=
1 / 15
x √(2 x 30 x 30 x 20 x 10)
=
1 / 15
x 30 x 20 = 2 x 20 = 40
Hence, the Perpendicular distance is 40 cm.

60. The two parallel sides of a trapezium of area 800 sq cm measure 25 cm and 55 cm. What is the height of the trapezium.
  A.  20 cm
  B.  35 cm
  C.  15 cm
  D.  25 cm
     
   
View Answer

Shortcut:
To find the height of the trapezium if its area and parallel sides are given.
Height =
2A / a + b
units
where, A = Area of trapezium, a and b are the length of parallel sides of the trapezium.

Here, A = 800, a = 25, b = 55
Using these values in the shortcut, we get:
Height =
2 x 800 / 25 + 55

=
2 x 800 / 80
= 2 x 10 = 20
Hence, the height is 20 cm.

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