c Use the Triangle Inequality Theorem to form three inequalities that must be true in order for the straws to form a triangle.
A
a Yes
B
b No
C
c 2/3 or 66 23 %
Practice makes perfect
a Let's start but cutting our straw in half to get two 3 -inch pieces. Now, let's place those pieces with the 4- inch straw to see if we can make a triangle.
The Triangle Inequality Theorem can help us verify that those lengths do form a triangle. We will need to compare the sum of two side lengths to the length of third side and repeat for the other combinations. To be able to form a triangle, the sum will need to be greater than the remaining side length.
Side
Sum of Other Two Sides
Inequalities
AB=4
3+ 3 = 6
4< 6
AC= 3
4 + 3= 7
3 < 7
BC= 3
4+ 3= 7
3 < 7
All the inequalities are true, therefore the three straw lengths can form a triangle.
b Let's follow a similar process as Part A to see if a triangle can be formed with sides length 1, 5, and 4 inches. Now, let's place those pieces with the 4-inch to see if we can make a triangle.
Just as in Part A, we will look at the sums of two side length and compare those sums to the length of the remaining side.
Side
Sum of Other Two Sides
Inequalities
AB=4
5+ 1 = 6
4< 6
AC= 1
4 + 5= 9
1 < 9
BC= 5
4+ 1= 5
5 ≮ 5
Not all of the inequalities hold meaning that we cannot form a triangle with these straw lengths.
c From Parts A and B, we know that some lengths will form triangles and some will not. Here, we are asked to generalize the situation and find the probability of being able to form a triangle by randomly cutting the 6 inch straw. If we cut a length of a off of the straw, the remaining length will be 6-a.
Let's apply the Triangle Inequality Theorem just as we did in Parts A and B comparing the sum of two side lengths with the length of the remaining side.
Side
Sum of Other Two Sides
Inequalities
AB=4
6-a+ a = 6
4< 6
AC= 6-a
4 + a= 4+a
6-a < 4+a
BC= a
6-a+4= 10-a
a < 10-a
In order for the lengths to form a triangle, all three of the inequalities must hold true. We see that the first is always true. Let's see when the other two hold true by isolating a in each of them.
ccc
6- a<4+ a&and& a<10- a
⇕&&⇕
a>1&and& a<5
Since 1
This means that there is a 4 inch section where the random cut could be made on the 6 inch straw that allows for the creation of a triangle. The probability of the random cut being in this section is the ratio of length of this section compared to the entire length of the straw.
Length of Cutting Zone/Length of Straw = 4in/6in=2/3
Therefore, the probability of getting two lengths that will form the other two sides of a triangle is 23 or 66 23 %.
