## PERIMETER OF A TRIANGLE

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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