Final answer:
The sides of the triangle are 10 units, 12 units, and 14 units, with the longest side being 14 units. We establish a system of equations based on the properties given and solve for the lengths of the consecutive even integer sides.
Step-by-step explanation:
To solve the problem of finding the lengths of the sides of a triangle where the sides are consecutive even integers and the longest side is 22 units shorter than the perimeter, we can establish a system of equations. Let's denote the shortest side as 'x', the middle side as 'x+2', and the longest side as 'x+4'. The perimeter of the triangle is the sum of its sides, so we have:
P = x + (x + 2) + (x + 4)
According to the problem, the longest side (x + 4) is 22 units shorter than the perimeter:
x + 4 = P - 22
Substituting the expression for the perimeter we get:
x + 4 = (x + x + 2 + x + 4) - 22
Simplifying this equation we find:
x + 4 = 3x + 6 - 22
x + 4 = 3x - 16
2x = 20
x = 10
Therefore, the sides of the triangle are 10 units, 12 units, and 14 units, with the longest side being 14 units.