the lengths of the triangle are consecutive odd integers, find the length of the shortest side if the perimeter is 19 units shorter than 4 times the lenghts of that side
let the sides be n-2,n,n+2
p = 3n = 4(n-2) - 19
3n = 4n - 8 - 19
-n = -27
n = 27
so, the sides are 25,27,29
81 = 4*25 - 19
Let's represent the length of the shortest side as x.
According to the problem, the lengths of the triangle are consecutive odd integers. So, we can say that the three sides of the triangle are x, x+2, and x+4.
Now, the perimeter of a triangle is the sum of the lengths of its sides. In this case, the perimeter is given to be 19 units shorter than 4 times the length of the shortest side.
So, we can write the equation as:
x + (x+2) + (x+4) = 4x - 19
Simplifying the equation, we get:
3x + 6 = 4x - 19
Bringing the x terms to one side and the constant terms to the other side, we have:
4x - 3x = 6 + 19
x = 25
Therefore, the length of the shortest side of the triangle is 25 units.