Using the Triangle Inequality Theorem, calculate the possible distance of the second route and then subtract the distance of the first route.
From 0 to 12 miles.
Practice makes perfect
Let's start with analyzing the given diagram. We are given the measures of two sides of the triangle.
To find the possible measure of the third side, let's recall what the Triangle Inequality Theorem states.
The sum of the lengths of any two
sides of a triangle must be greater
than the length of the third side.Let the distance between Aisha's house and the park along Main Street be y miles. Now we can use the above theorem to form three inequalities true for the triangle on the diagram.
6+7.5>y
6+y>7.5
7.5+y>6
We will solve these inequalities by isolating y on one side of each inequality.
Inequality
Solution Set
6+7.5>y
13.5>y
6+y>7.5
y>1.5
7.5+y>6
y>- 1.5
Let's graph these solution sets and find the common solutions.
We can see that all three lines overlap on the segment from 1.5 to 13.5. We conclude that the distance from Aisha's house to the park along Main Street is between 1.5 to 13.5 miles. If Aisha's riding to the park along Route 3 and Clay Road, she will ride the sum of 7.5 and 6 miles.
7.5+6=13.5miles
The maximum distance of the second route is 13.5 miles. In this case, the additional distance is the following.
13.5-13.5=0miles
The minimum distance of the second route is 1.5 miles. Subtracting this value from 13.5, we can calculate the additional distance Aisha would need to ride.
13.5-1.5=12miles
Thus, the additional distance is between 0 and 12 miles.
