PERIMETER OF PARALLELOGRAM

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 = √[(x₂ - x₁)² + (x₂ - x₁)²]

Length of AB :

Substitute (x₁, y₁) = A(5, 8) and (x₂, y₂) = 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 (x₁, y₁) = B(6, 3) and (x₂, y₂) = 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 (x₁, y₁) = D(2, 6) and (x₂, y₂) = 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 (x₁, y₁) = A(5, 8) and (x₂, y₂) = 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

Example 5 :

If the sides of a parallelogram are increased to twice its original lengths, how much will the perimeter of the new parallelogram?

(a) 1.5 times     (b) 2 times    (c) 3 times    (d) 4 times

Solution :

Let a and b be the side lengths of parallelogram. 

Perimeter of parallelogram = 2(a + b)

New dimension of parallelogram are 2a and 2b

Perimeter of new parallelogram = 2(2a + 2b)

= 2(2) (a + b)

= 2(perimeter of old parallelogram)

Perimeter of new parallelogram is two times the perimeter of old parallelogram.

Example 6 :

The vertices of a rectangle are F (1, 6), G (7, 6), H (7, 2), and J (1, 2). Draw the rectangle in a coordinate plane and find its perimeter.

Solution :

Length of rectangle = 7 - 1

= 6 units

Width of rectangle = 6 - 2

= 4 units 

Perimeter of rectangle = 2(length + width)

= 2(6 + 4)

= 2(10)

= 20 units.

So, the perimeter of the rectangle is 20 units.

Example 7 :

What is the perimeter of the rectangle with the vertices shown below? 

A (4, 7), B (4, 15), C (9, 15), D (9,7)

a) 8 units     b) 13 units     c) 26 units    d) 70 units

Solution :

By drawing the rectangle with the given vertices, we get the length of rectangle as AB and width of rectangle as BC.

Length of rectangle = 15 - 7

= 8 units

Width of rectangle = 9 - 4

= 5 units

Perimeter of rectangle = 2 (length + width)

= 2(8 + 5)

= 2 (13)

= 26 units

So, the perimeter of the rectangle is 26 units.

Example 8 :

find the area and perimeter of parallelogram ABCD if the area of shaded triangle is 9 cm².

Solution :

Area of triangle = (1/2) x base x height

Let h be the height of the triangle.

(1/2) x 3 x h = 9

h = 9(2/3)

= 6 cm

So, height of the triangle is 6 cm.

In the triangle ABE,

AB² = AE² + BE²

AB² = 6² + 3²

= 36 + 9

= 45

AB = √45

= √(3 x 3 x 5)

= 3√5

Perimeter of triangle = 2(3 + 4 + 3√5)

= 2(7 + 3√5)

= 14 + 6√5 cm

Area of parallelogram = base x height

= 7 x 6

= 42 cm²

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