If line segment AB is A(10, 4) B(6,-1) and point P is 3/8 alomg line segment AB, what are ther coordinates of?

Question

(graphing question) I've tried adding the y snd x of points A and B and multipying the result by 3/8, but it won't work! How do i do it?? ( to get P if P in exactly 1/2 along lin AB, you add them together and divide by 2-multiplying by 1/2 also works.)

Answer 1

When you say that P is 3/8 of the way along AB, I assume you mean it's closer to A than B.

Method 1: Convex combination.
Fancy name for a simple idea. As you said, if P is half way, you just average the endpoints. In this case, you'll do a *weighted* average. Add 5 times A to 3 times B, divide by 8. This gives (8.5, 2.125).

A little more background: imagine breaking AB into 8 equal parts. Start at A and count off 3 parts to reach P; or, start at B and count off 5 parts. You could think of A and B as points having a tug-of-war with P at the middle of the rope. Since P is closer to A, A pulls harder. Each unit of distance from one vertex counts as a unit of "pull" for the other; that is, A pulls with a strength of 5 and B pulls with a strength of 3. A counts 5 times, B counts 3 times. If you add up 5A and 3B, though, that's 8 points. Divide by 8 to finish. (The same principle applies when computing your quiz grade if you got 10 points on each of 5 quizzes and 6 points on each of 3 quizzes.)


Method 2: Compute the difference.
When walking from A to B, you would go 4 steps left and 5 steps down. We say that the "directed distance" from A to B is (-4, -5). (If you prefer the word "vector", use that.) But P is not at the end of that path; it's 3/8 of the way. So, we start at A and go 3/8 of (-4,-5). We get to (10, 4) + (3/8)(-4,-5) = (10, 4) + (-1.5, -1.875) = (8.5, 2.125).

Answer 2

Consider that point X is the midpoint of AB ie ½ along line segment AB
now consider point Y is the midpoint of AX ie ½ along line segment AX and ¼ along AB
and finally consider point P as the midpoint of YX ie ½ along line segment YX and 3/8 along line segment AB.

A___________Y_____P_____X____________________B|


Now converting that into math:

Midpoint AB = X = [ (10/2 + 6/2), (4/2 + -1/2) ]
Coordinate of X = (8,1.5)

Midpoint AX = Y = [ (10/2 + 8/2), (4/2 + 1.5/2)]
Coordinate of Y = (9,2.75)

Midpoint of YX = P = [(8/2 + 9/2), (1.5/2 + 2.75/2)]
Coordinate of P = (8.5, 2.125)