Define variables, write an equation and solve. The perimeter of a scalene triangle is 75 inches. The middle side is 6 inches longer than the shortest side. The longest side is 15 inches less than twice the middle side. Find the length of all three sides.
s +(s+6) + 2(s+6)-15 = 75
To solve this problem, let's define the variables:
Let x be the length of the shortest side.
Let y be the length of the middle side.
Let z be the length of the longest side.
Now let's translate the given information into equations:
1. The perimeter of a triangle is the sum of the lengths of its sides. In this case, the perimeter is given as 75 inches. So we can write the equation:
x + y + z = 75 ...(Equation 1)
2. We are given that the middle side is 6 inches longer than the shortest side. Therefore:
y = x + 6 ...(Equation 2)
3. We are also given that the longest side is 15 inches less than twice the middle side. Therefore:
z = (2y) - 15 ...(Equation 3)
Now we have a system of three equations (Equations 1, 2, and 3) with three variables (x, y, and z). We can solve this system of equations to find the values of x, y, and z.
To solve the system, we can substitute the value of y from Equation 2 into Equations 1 and 3. Let's do that:
Substituting y = x + 6 into Equation 1:
x + (x + 6) + z = 75
2x + z = 69 ...(Equation 4)
Substituting y = x + 6 into Equation 3:
z = (2(x + 6)) - 15
z = 2x + 12 - 15
z = 2x - 3 ...(Equation 5)
Now we have a system of two equations with two variables (Equations 4 and 5). We can solve this system to find the values of x and z.
Let's solve this system using the method of substitution:
From Equation 4, z = 69 - 2x
Substituting this value of z into Equation 5:
2x - 3 = 69 - 2x
4x = 72
x = 18
Now that we have the value of x, we can substitute it back into Equation 2 to find y:
y = x + 6
y = 18 + 6
y = 24
Finally, we can substitute the values of x and y into Equation 3 to find z:
z = 2x - 3
z = 2 * 18 - 3
z = 33
Therefore, the lengths of the three sides of the scalene triangle are:
Shortest side (x) = 18 inches
Middle side (y) = 24 inches
Longest side (z) = 33 inches