Question 1189872: This question is really all about Distance and Midpoint Formula
You are going on a two-day hike. The map at the right shows the trails you plan to follow. (Each unit represents 1 mile.)
a. You hike from the lodge to point A and decide that you will hike to the midpoint of AB before you camp for the night. At what point in the plane will you be camping?
b. How far will you hike each day?
I think I can’t post the illustration but I hope opening this link would help https://mrrossatgradyhigh.files.wordpress.com/2016/08/10-1-distance-and-midpoint-formulas.pdf
(Located 4th page of the pdf, question number 16)
Answering this question would help me a lot, thank you!
Found 2 solutions by Boreal, Theo:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The midpoint of AB has coordinate x which is the average of the two x components (-0.50) and the same for the two y components (4), so it is at (-0.5, 4).
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first day is sqrt(13) distance because distance is sqrt(difference x^2+difference y^2). That is to A.
Easier to do AB, and that distance is sqrt(5^2+4^2)=sqrt(41). Unlike slope, where which order you use the variable matters, it doesn't matter with distance, since you are squaring. You hike half of that the first day, so the total distance is sqrt(13)+(0.5)sqrt(41). That can be put into decimal form.
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The total distance is sqrt(13)+sqrt(41)+ the last leg, which is sqrt(40). The distance the second day is
0.5*sqrt(41)+ sqrt (40)
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the midpoint from point A to point B would be found by the formula:
midpoint = ((x1 + x2)/2, (y1 + y2)/2)
(x1,y1) = (-3,2)
(x2,y2) = (2,6)
midpoint is equal to ((-3 + 2)/2, (2 + 6)/2) which is equal to (-1/2,4).
this can also be shown as (-.5,4).
you will be camping at the point (-1/2, 4).
we'll call that point C.
point C is the midpoint between point A and point B.
on the first day, you will have hiked from the lodge to point A and then to point C.
the distance between the lodge and point A is equal to sqrt((-3)^2 + 2^2) = sqrt(9 + 4) = sqrt(13).
point A is at (-3,2).
point C is at (-.5,4).
-.5 - (-3) is equal to 2.5
4 - 2 is equal to 2
the distance between point A and point C is equal to sqrt(2.5^2 + 2^2) which is equal to sqrt(6.25 + 4) which is equal to sqrt(10.25).
the distance traveled from the lodge to point A and then to point C is therefore equal to sqrt13) + sqrt(10.25) = 6.807113394 kilometers.
if you continue in the same direction you were traveling to complete the loop from the lodge to point A to point B to the lodge, then you will be traveling from point C to point B to the lodge on the second day.
the distance from point C to point B is the same as the distance from point A to point C, which is equal to sqrt(10.25).
the distance between point B and the lodge is equal to sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40).
the total distance you travel on the second day must be sqrt(10.25) + sqrt(40) = 9.526117439 kilometers.
your solution is that you stopped at the point (-.5,4) at the end of the first day and that you traveled 6.807113394 kilometers on the first day and 9.526117439 kilometers on the second day.
let me know if you have any questions.
theo
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