Anonymous

# Division of Line Segment?

I was studying for my math exam can you help me answer this for my review in my exam next week.... i just find it difficult so can you help me..... pls.......... help me..........

A point P(x,y) is on the line through A(-4, 4) and B(5,2).

a)The coordinates of P given that the segment AB is extended throught B to P so that P is twice as far from B, and

b) The coordinates of P given that AB is extended through A to P so that P is three times as far from B as from A.

Relevance

For a): I presume you mean that Point P is twice as far from A as from B. If this is so, then Point B becomes the midpoint of line segment ABP. Label the coordinates as (x_A, y_A) for A, (x_B,Y_B) for B, and (x_P,y_P) for P. Then by the midpoint formula:

x_B = (x_A + x_P)/2 (Eq1)

and

y_B = (y_A + y_P)/2. (Eq2)

Since A has coordinates (-4,4) and B has coordinates (5,2) then by Eq1

5 = (-4 + x_P)/2 (Eq3)

and by Eq2

2 = (4 + y_P)/2. (Eq4)

Solving Eq3 gives x_P = 14; Eq4 gives y_P = 0.

Hence, the coordinates of P are (14,0). So the line is determined by the points A(-4,4), B(5,2) and P(14,0).

For b): Point A becomes the point one-third (1/3) the way from Point P to Point B, i.e. |PA| = (1/3) |PB| or |PB|=3|PA|. The general formula for the point M(x_M,y_M) in between two points, say C(x_1,y_1) and D(x_2,y_2), that is a fraction p of the distance from C to D has coordinates

x_M = (1-p)x_C + px_D

and

y_M = (1-p)y_C + py_D.

In the given problem, Point P is C, B is D, and A is M, i.e.,

C---M------D

P---A------B.

Hence,

x_A = (1-p)x_P + px_B

and

y_A = (1-p)y_P + py_B.

Plugging-in the given p=1/3, A(-4,4), and B(5,2) in the preceding equations give

-4 = (2/3)x_p + (1/3)*5 (Eq5)

and

4 = (2/3)y_P + (1/3)*2. (Eq6)

Solving Eq5 and Eq6 for x_P and y_P, respectively, gives

x_P = ((-4)(3) - 5)/2 = -17/2 and

y_P = ((4)(3) - 2)/2 = 5.

Hence, the desired coordinates for P are (-17/2 , 5).