Answer:
The lengths of each side are:
a = 4 cm.
b = 7 cm.
c = 8 cm.
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