Draw diagrams with different positions of the houses.
Between 1 and 5 miles.
Practice makes perfect
Let's draw a triangle with High Street, Main Street and 5th Street as three sides, the houses of Leonard and Josh as two vertices L and J, and the corner of Main Street and 5th Street as the third vertex C. Depending on the angle between Main Street and 5th Street, we get different triangles.
It is given that LC=2 miles and CJ=3 miles. We are asked to find the range of possible values of JL.
Since LC, CJ, and JL are three sides of a triangle, we can use the Triangle Inequality Theorem to get bounds for JL.
The sum of the length of any two sides of a triangle
is greater than the length of the third side.
The key word here is any. This theorem gives a restriction on the length of the sides in three ways.
Inequality
Consequence
LC+ CJ&> JL 2+ 3&> JL
JL< 2+ 3=5
CJ+ JL&> LC 3+ JL&> 2
Always true
JL+ LC&> CJ JL+ 2&> 3
JL> 3- 2=1
Notice that since 3> 2, the second inequality is always true. The first inequality tells us that JL cannot be too long, and the third inequality tells that JL cannot be too short.
1< JL<5
The distance between Leonard and Josh's house on High Street is somewhere between 1 and 5 miles.
