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.
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.
sides are n, n+2, n+4
n+4+22=3n+6
check that.