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).

**T****he shortest path is sqrt8+sqrt20.**

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**

## See eNotes Ad-Free

Start your **48-hour free trial** to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Already a member? Log in here.