Start with finding the greatest possible length of West-East walkway by the Triangle Inequality Theorem. Notice that the length of West-East does not need to be a whole number.
129 yd
Practice makes perfect
Given figure shows the walkways connecting four dormitories on a college campus.
We will find the greatest possible whole number length for the walkway between South Dorm and East Dorm. In order to that we will first find the greatest possible length of West-East walkway by using Triangle Inequality Theorem. This theorem says that:
The sum of the lengths of any two sides
of a triangle is greater than the
length of the third side.
The illustration of the theorem is given below.
XY+YZ>XZ
YZ+XZ>XY
XZ+XY>YZ
Applying the theorem, we have three inequalities.
Inequality I:& 42+57>WE ⇒ 99>WE
Inequality II:& 57+WE>42 ⇒ WE > -15
Inequality III:& WE+42>57 ⇒ WE > 15
Combining these inequalities, we can determine the range of the possible lengths of WE.
Combined Inequality: 15< WE < 99
Notice that the length of WE does not need to be whole number. Therefore, the greatest length of WE can be 98.9.... Next, we will use the Triangle Inequality Theorem one more time to determine the length of South-East walkway.
Inequality I:& 31+98.9>SE ⇒ 129.9>SE
Inequality II:& 98.9+SE>31 ⇒ SE > -67.9
Inequality III:& SE+31>98.9 ⇒ SE > 67.9
Let's combine the inequalities!
Combined Inequality: 67.9< SE < 129.9
As a result, the greatest possible whole number length of South-East walkway is 129 yd.
