Explanation:
We can use the Triangle Inequality theorem to determine the range of the possible distances between Leonard and Josh's houses when traveling straight down High Street.
According to the Triangle Inequality theorem, for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. We can use this property to determine the possible range of distances.
Let's assume that "a" is the distance between Leonard's house and the intersection of Main Street and 5th Street, "b" is the distance between Josh's house and the intersection of Main Street and 5th Street, and "c" is the distance between Leonard's house and Josh's house when traveling straight down High Street.
Using the Pythagorean theorem, we can find that:
a^2 + b^2 = (3 + 2)^2 = 25
We can also use the Triangle Inequality theorem to find that:
c < a + b c > |a - b|
Substituting the values for "a" and "b," we get:
c < sqrt(25) = 5 c > |a - b| = |sqrt(25 - b^2) - sqrt(25 - a^2)|
To find the maximum possible value of "c," we want to minimize the expression for "c". This occurs when "a" and "b" are as close together as possible, which happens when "a" = "b".
Substituting "a" = "b" into the first equation, we get:
2a^2 = 25 a^2 = 12.5 a = b ≈ 3.54
Substituting these values into the expression for "c," we get:
c > |3.54 - 3.54| = 0
Therefore, the maximum possible distance between Leonard and Josh's houses when traveling straight down High Street is 0, which means they are in the same location.
To find the minimum possible value of "c," we want to maximize the expression for "c." This occurs when "a" and "b" are as far apart as possible, which happens when one of them is 0.
If "a" = 0, then:
b^2 = 25 b ≈ 5
Substituting these values into the expression for "c," we get:
c < 3.54 + 5 ≈ 8.54