You have two pieces of wood that will make up two sides of a triangular picture frame. One is 8 in. long and the other is 11 in. long. What is the range of possible lengths for the third side of the frame? Please show all work in order to get full credit for this problem.
To determine the range of possible lengths for the third side of a triangular picture frame, we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference of the lengths of any two sides of a triangle must be less than the length of the third side.
Let's call the lengths of the two sides a and b, and the length of the third side c. We have:
a = 8 inches
b = 11 inches
c = unknown
According to the Triangle Inequality Theorem, these inequalities must hold:
1. a + b > c
2. a + c > b
3. b + c > a
We can use these inequalities to find the range for c. Let's start with inequality 1:
8 + 11 > c
19 > c
This means that the third side must be less than 19 inches long.
Next, let's look at inequality 3, as inequality 2 will always be true since a < b:
11 + c > 8
c > 8 - 11
c > -3
A negative length does not make sense in this context, so the minimum positive length that c could be (to construct a triangle) is when c is just slightly larger than the difference between b and a. Therefore, c must be greater than the difference between 11 inches and 8 inches:
c > 11 - 8
c > 3 inches
So combining the two constraints, we can say:
3 inches < c < 19 inches
This means that to construct a triangle, the length of the third side c must be greater than 3 inches and less than 19 inches. This is the range of possible lengths for the third side of the triangular picture frame.