The perimeter of a triangle is 33 inches. The sum of the length of the longest side and twice the length of the
shortest side is 31 inches. Twice the length of the longest side minus the sum of the lengths of the other sides is
12 inches. Find the lengths of all sides.
How do I set it up and write out my equations?
shortest side --- x
middle size ----- y
largest side ----- z
first equation is obvious: x+y+z = 33 , #1
second equation:
"The sum of the length of the longest side and twice the length of the shortest side is 31 inches"
---> z + 2x = 31 , #2
third equation:
"Twice the length of the longest side minus the sum of the lengths of the other sides is
12 inches"
--->
2z - (x+y) = 12
2z - x - y = 12 , #3
add #1 and #2
3z = 45
z = 15
plug into #2:
15 + 2x = 31
2x = 16
x = 8
plug the two values found into #1
x+y+z= 31
8+y+15 = 33
y = 10
the sides are 8, 10, and 15
check my arithmetic
To set up and write out the equations for this problem, let's assign variables to the lengths of the three sides of the triangle.
Let's say the lengths of the sides are a, b, and c.
Now, let's write down the given information as equations:
1. The perimeter of a triangle is 33 inches. This means that the sum of the lengths of all three sides is equal to 33. So, we can write the equation: a + b + c = 33.
2. The sum of the length of the longest side and twice the length of the shortest side is 31 inches. Let's assume that 'c' is the longest side and 'a' is the shortest side. So, the equation can be written as: c + 2a = 31.
3. Twice the length of the longest side minus the sum of the lengths of the other sides is 12 inches. Using the same assumptions, the equation can be written as: 2c - (a + b) = 12.
With these three equations, we have a system of equations that can be solved simultaneously to find the lengths of all sides of the triangle.