The lengths of the sides of a triangle are consecutive even integers. Find the length of the longest side if it is 22 units shorter of the perimeter.
sides are n, n+2, n+4
n+4+22=3n+6
check that.
To solve this problem, let's assume that the three consecutive even integers representing the lengths of the sides of the triangle are x, x+2, and x+4 (since consecutive even integers have a difference of 2 units between them).
We are given that the longest side is 22 units shorter than the perimeter. The perimeter of a triangle is calculated by adding up the lengths of all its sides. So, the perimeter is x + (x+2) + (x+4).
According to the given information, the longest side (x+4) is 22 units shorter than the perimeter. This can be expressed as:
x+4 = (x + (x+2) + (x+4)) - 22
Now we can solve for x:
x+4 = 3x + 6 - 22
Simplifying this equation, we have:
x+4 = 3x - 16
Subtracting x from both sides:
4 = 2x - 16
Adding 16 to both sides:
20 = 2x
Dividing both sides by 2:
x = 10
Now that we have found x, we can find the lengths of the sides of the triangle:
The three consecutive even integers are:
x = 10
x+2 = 10+2 = 12
x+4 = 10+4 = 14
Therefore, the lengths of the sides of the triangle are 10, 12, and 14 units. The longest side is 14 units long.