Coordinate geometry - Midpoint of a line segment

Question

- 5. A ship is 80 km west and 40 km north of a small island which is 20 km west and 20 km north of a port. A yacht is midway between the island and the ship. Give the position of the yacht

a. relative to the island
b. relative to the port.

*I don't understand a. and b. by "relative to island and port."

 

 

- 6. Find the distance between the midpoint of the line segment joining (–1,–2)

and (1,4) and the midpoint of the line segment joining (3,14) and (7,12).

*This question is confusing to understand

 

- 7. Photo - http://imgur.com/NDtu2W3

 

*Not sure how to answer this question. 

 

Any answers is appreciated and thanks for using your time to answer these questions.

 

Answer 1

Davado,

5) is basically asking you to find the coordinates of the yacht from two different origins, the island and the port. You need to pick one to be your origin, then find the coordinates of the others based on that. I would choose the port for the origin, so PORT (0, 0)

 

The island is 20 west and 20 north of the island. ISLAND (-20, 20)

 

The ship is 80 west and 40 north of the island. SHIP (-100, 60)

 

The yacht is on the midpoint between the island and the ship.  (-120/2, 80/2)   YACHT  (-60, 40)

 

Now since we don't have actual coordinates for any of it, we express the location of the yacht as the change in x and y from the desired reference:

 

a)  (x-40, y+20)  or (W40, N20)

 

b)  (x-60, y+40) or (W60, N40)

 

6) This just requires you to find the midpoint of the two lines and then find the distance between those two midpoints. Remember that a midpoint is found by simply taking the average of the x's and the average of the y's.

 

(-1 + 1)/2 = 0    (-2 + 4)/2 = 1      M.P.  (0, 1)

(3 + 7)/2 = 5     (14 + 12)/2 = 13     M.P.    (5, 13)

 

The distance between them is found by the distance formula, which is just a derivation of Pythagorean theorem:

 

d = √[(5 - 0)² + (13 - 1)²]

d = √[(5)² + (12)²]

d = √[25 + 144]

d = √[169]

d = 13

 

7) This requires you to find the general coordinates of M and N based on the midpoint formula, and then find the distance MN and the distance BC.

 

M ⇒ [(x₁ + x₂)/2, (y₁ + y₂)/2]         N ⇒ [(x₁ + x₃)/2, (y₁ + y₃)/2]

 

BC = √[(x₃ - x₂)² + (y₃ - y₁)²]

 

MN = √{[(x₁ + x₃)/2 - (x₁ + x₂)/2]² + [(y₁ + y₃)/2 - (y₁ + y₂)/2]²}

MN = √{[(x₃ - x₂)² + (y₃ - y₂)²]/4}

MN = √[(x₃ - x₂)² + (y₃ - y₂)²]/2 

 

so by substitution:

 

MN = BC/2

 

The rational expressions are hard to show with this editor. I hope this helps.