The perimeter of a triangle is 19 cm. Assume the sides (in order of length from smallest to longest are a, b, c respectively). If the length of the longest side is twice that of the shortest and is 3 cm less than the sum of the lengths of the other two sides, find the lengths of each side.

The lengths of each side are:

a = 4 cm.

b = 7 cm.

c = 8 cm.                

Step-by-step explanation:

To solve this, we will write the equation out of the word problem given in the question, and this is done as follows;

sides of the triangle from smallest to longest = a, b, c

perimeter of the triangle = a + b + c = 19 cm - - - - - - - - - (1)

Length of the longest side is twice that of shortest side; c = 2a - - - - - - (2)

length of longest side is equal to 3 cm less than the sum of the lengths of the other two sides; c = (a + b) - 3  - - - - - - - (3)

therefore:

a + b + c = 19 - - - - - (1)

c = 2a - - - - - - - - - - (2)

c = (a + b) - 3 - - - - - - - - (3)

putting equation (2) into equation (1)

a + b + (2a) = 19

3a + b = 19 - - - - - - (4)

Substituting for c in equation (3) using equation (2)

c = (a + b) - 3 - - - - - - - - (3)  (replacing c with 2a from equation 2)

2a = a + b - 3 (making b the subject)

2a - a = b - 3

∴ b = 2a - a + 3

b = a + 3 - - - - - - -(5)

substituting for the value of b in equation (4), using equation 5

3a + b = 19 - - - - - - (4)  ( replacing b with equation 5)

3a + (a + 3) = 19

4a + 3 = 19

4a = 19 - 3 = 16

∴ a = 16 ÷ 4 = 4

since, we know 'a', calculating for b, using equation 5;

b = a + 3 - - - - - - -(5) (where a = 4)

∴ b = 4 + 3 = 7

Finally calculating for c, using equation 2

c = 2a - - - - - - - (2) (where a = 4)

c = 2 × 4 = 8

Therefore:

a = 4; b = 7 ; c = 8