Perimeter is a path that surrounds a two dimensional shape. The term may be used either for the path or its length it can be thought of as the length of the outline of a shape.
Parallelogram is a quadrilateral in which opposite sides are parallel and equal as shown below.
Formula for perimeter of a parallelogram :
= 2a + 2b
= 2(a + b) units
Example 1 :
Find the perimeter of a parallelogram with the base and side length are 15 cm and 12 cm.
Solution :
Perimeter of a parallelogram :
= 2(a + b)
Substitute a = 15 and b = 12.
= 2(15 + 12)
= 2(27)
= 54 cm
Example 2 :
Find the perimeter of parallelogram with the base and side length are 9 ft and 3 ft.
Solution :
Perimeter of a parallelogram :
= 2(a + b)
Substitute a = 9 and b = 3.
= 2(9 + 3)
= 2(12)
= 24 cm
Example 3 :
If the perimeter of parallelogram is 40 in and its base is 15 in, find its side length.
Solution :
Perimeter of a parallelogram = 40
2(a + b) = 40
Substitute a = 15.
2(15 + b) = 40
30 + 2b = 40
Subtract 30 from each side.
2b = 10
Divide each side by 2.
b = 5
The side length of the parallelogram is 5 in.
Example 4 :
Prove that the points (5, 8), (6, 3), (3, 1) and (2, 6) form a parallelogram and also find the perimeter.
Solution :
Let A(5, 8), B(6, 3), C(3, 1) and D(2, 6).
Formula to find the distance between two points :
d = √[(x_{2} - x_{1})^{2} + (x_{2} - x_{1})^{2}]
Length of AB :
Substitute (x_{1}, y_{1}) = A(5, 8) and (x_{2}, y_{2}) = B(6, 3) in the distance formula.
AB = √[(6 - 5)^{2} + (3 - 8)^{2}]
= √[(1)^{2} + (-5)^{2}]
= √[1 + 25]
= √26
Length of BC :
Substitute (x_{1}, y_{1}) = B(6, 3) and (x_{2}, y_{2}) = C(3, 1) in the distance formula.
BC = √[(3 - 6)^{2} + (1 - 3)^{2}]
= √[(-3)^{2} + (-2)^{2}]
= √[9 + 4]
= = √13
Length of DC :
Substitute (x_{1}, y_{1}) = D(2, 6) and (x_{2}, y_{2}) = C(3, 1) in the distance formula.
DC = √[(3 - 2)^{2} + (1 - 6)^{2}]
= √[(1)^{2} + (-5)^{2}]
= √[1 + 25]
= = √26
Length of AD :
Substitute (x_{1}, y_{1}) = A(5, 8) and (x_{2}, y_{2}) = D(2, 6) in the distance formula.
AD = √[(2 - 5)^{2} + (6 - 8)^{2}]
= √[(-3)^{2} + (-2)^{2}]
= √[9 + 4]
= = √13
length of AB = length of DC
length of AD = length of BC
Opposite sides are equal. So, the given points form a parallelogram.
Perimeter = 2(√26 + √13) units
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