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.
Let a, b and c be the side lengths of a triangle.
Perimeter of Triangle = a + b + c
Example 1 :
Find the perimeter of the triangle shown below.
Solution :
Perimeter of the triangle :
= 9 + 9 + 4
= 22 in
Example 2 :
The sides of a triangle are 12 cm, 6 cm and 8 cm. Find the perimeter of the triangle.
Solution :
Perimeter of the triangle :
= 12 + 6 + 8
= 26 cm
Example 3 :
The length of each of side of an equilateral triangle is 7 in. Find the perimeter of the triangle.
Solution :
Perimeter of the triangle :
= 7 + 7 + 7
= 21 in
Example 4 :
The length of each of the equal sides of an isosceles triangle is 3 cm and the length of the third side is 5 cm. Find the perimeter of the triangle.
Solution :
Perimeter of the triangle :
= 3 + 3 + 5
= 11 cm
Example 5 :
The sides of a triangle are 2x, 3x and 4x. If the perimeter of the triangle is 81 in, then find the length of each side of the triangle.
Solution :
Perimeter of the triangle = 81 in
2x + 3x + 4x = 81
9x = 81
Divide each side by 9.
x = 9
2x = 2(9) = 18
3x = 3(9) = 27
4x = 4(9) = 36
The lengths of the sides of the triangle are 18in, 27 in and 36 in.
Example 6 :
If the perimeter of the triangle shown below is 101 cm, find the value of x.
Solution :
The above triangle is an isosceles triangle and the length of each of the two equal sides is (x + 3) cm.
Perimeter of the triangle = 101 cm
(x + 3) + (x + 3) + (x + 5) = 101
x + 3 + x + 3 + x + 5 = 101
3x + 8 = 101
Subtract 8 from each side.
3x = 93
Divide each side by 3.
x = 31
Example 7 :
The length of each of the equal sides of an isosceles triangle is 4 cm longer than the base. If the perimeter of the triangle is 62 cm, find the length of the sides of the triangles.
Solution :
Let x be the length of each of the two equal sides and y be the length of the base.
Then, x = y + 4.
Perimeter of the triangle = 62 cm
x + x + y = 62
2x + y = 62
Substitute x = y + 4.
2(y + 4) + y = 62
2y + 8 + y = 62
3y + 8 = 62
Subtract 8 from each side.
3y = 54
Divide each side by 3.
y = 18
x = 18 + 4
= 22
The length of the sides of the triangle are 22 cm, 22 cm and 18 cm.
Example 8 :
Find the perimeter of the triangle defined by the following points :
D(1, 3), E(8, 3) and F(4, 7)
Solution :
Plot the points in the coordinate plane. Draw the height from F to the side DE. Label the point where the height meets DE as G. Point G has coordinates (4, 3).
Length of DE :
= 8 - 1
= 7
Use distance formula to find the length of EF and FD.
Distance Formula :
= √[(x2 - x1)2 + (y2 - y1)2]
Length of EF :
(x1, y1) = E(8, 3)
(x2, y2) = F(4, 7)
EF = √[(4 - 8)2 + (7 - 3)2]
EF = √[(-4)2 + (4)2]
EF = √[16 + 16]
EF = √32
EF = √(4 ⋅ 4 ⋅ 2)
EF = 4√2
Length of FD :
(x1, y1) = F(4, 7)
(x2, y2) = D(1, 3)
FD = √[(1 - 4)2 + (3 - 7)2]
FD = √[(-3)2 + (-4)2]
FD = √[9 + 16]
FD = √25
FD = 5
Perimeter of the triangle :
= 5 + 7 + 4√2
= (12 + 4√2) units
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