A map of Jane's town with her home and workplace is sown. Jane wants to determine the shortest route from her home to her workplace. She walks only on the sidewalks indicated by dotted lines on the map. What is the distance of the shortest route, to the nearest whole block

Jane's shortest route from her home to her workplace is the route that passes along the block in the middle of the graph, which is 2.5 blocks shorter, than the apparently next shortest route.

How to calculate the distance between two points

The shortest distance between two points is a straight line.

The shortest route to her workplace is the route that approximates a straight line the most.

According to the dotted lines, there are two routes, that are approximately direct and therefore short routes to her workplace are;

Taking the route that passes close to the center of the map to her workplace

The route heading past the block close to the center of the map to her workplace
The route to the left of the route

Use the Pythagoras theorem to calculate the path.

For the first path

D= √(2² + 4²)

D = 2√5

For the second path

D2 = √(1² + 2²)

D2 = √5

For the third path

Path 3 = 1 + 1 = 2

For the fourth path

D4 = √(2² + 3²) = √13

For the fifth path

D5 = 3

Total distance path = 2√5 + √5 + 2 + √13 + 3

Total distance path ≈ 15.3

Taking the route to the left of the above route gives;

Path 1 = √(2² + 4²) = 2·√5

Path 2 = 7 + 6 = 13

Cumulative distance = 2·√5 + 13 ≈ 17.5

Hence the shortest route is the route that heads to her workplace = 17.5 blocks - 15.3 blocks = 2.5 blocks shorter

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