What is the shortest path between the three points (4, 7), (2, 3) and (0, 1)?  

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hala718 eNotes educator | Certified Educator

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To find the shortest path,we will calculate the length of the path between all three points.

The distance Between (4,7) and (2,3)

==> D1 = sqrt( 4-2)^2 + (7-3)^2 = sqrt(4+16) = sqrt20

The distance between (4,7) and ( 0,1)

==> D2 = sqrt(4^2 + (7-1)^2 = sqrt(16+36) = sqrt(52)

The distance between ( 2,3) and (0,1)

==> D3 = sqrt(2^2 + (3-1)^2 = sqrt(4+4) = sqrt8

Now we can conclude that the shortest distance is sqrts20.

Then we should go from (0,1) to (2,3) to (4,7).

The shortest path is sqrt8+sqrt20.

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justaguide eNotes educator | Certified Educator

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You want the length of the shortest path which has all the three points (4, 7), (2, 3) and (0, 1) on it.

Let us first find the distance between each of the points. We use the relation for the distance between tow points (x1, y1) and (x2, y2) as sqrt [(x1 – x2) ^2 + (y1 – y2) ^2].

(4, 7) and (2, 3): sqrt [(4 – 2) ^2 + (7 – 3) ^2] = sqrt [4 + 16] = sqrt 20

(4, 7) and (0, 1): sqrt [(4 – 0) ^2 + (7 – 1) ^2] = sqrt [16 + 36] = sqrt 52

(2, 3) and (0, 1): sqrt [(2 – 0) ^2 + (3 – 1) ^2] = sqrt [4 + 4] = sqrt 8

Therefore the shortest path would be if you go from (0, 1) to (2, 3) and then to (4, 7) and the total distance to be covered would be sqrt 8 + sqrt 20

The required result is sqrt 8 + sqrt 20

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