Answer this please: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(4, −2), B(9, 8); 2 to 3

Answer 1

Hello Makena,

I would ask you to first draw out the coordinate plane and plot the points A (4,-2) and B (9,8). With those 2 points, construct a right triangle where segment AB is the hypotenuse. Label the lengths of the other 2 sides by counting how long they are. You should get the vertical segment to be 10 and the horizontal segment to be 5.

Knowing that we will want to split the length of the hypotenuse into 5 segments (the ratio 2 to 3 means 2 parts to 3 parts, meaning we need a total of 5 parts), we can split each side length into 5 parts. The vertical segment will be split into groups of 2 (10 divided by 5), and the horizontal can be split into groups of 1 (5 divided by 5).

We can bring these measurements together onto the hypotenuse. Segment AP will be will be in the ratio of 2 groups. Horizontally we will go 2 units to the right (2 x 1) from point A and 4 units vertical (2 x 2). The resultant point is (6,2)

Answer 2

The formula is:

( (m*x2+n*x1)/(m+n),(m*y2+n*y1)/(m+n) )

(m*x2+n*x1) (m*y2+n*y1)

-------------- ---------------

m+n m+n

(x1,y1) = (4,-2)

(x2,y2) = (9,8)

m=2 and n=3

So

m+n = 5

(2*9 + 3*4)/5 = (18 + 12)/5 = 30/5 = 6

(2*8 + 3*-2)/5 = (16 + -6)/5 = 10/5 = 2

so P=(6,2) is 2/5 the distance from A=(4,-2) to B=(9,8)

============================================

CHECK: using distance formula

the AP = sqrt( 4 + 16) = sqrt(20) = 2 * sqrt(5)

PB = sqrt ( 9 + 36) = sqrt(45) = 3 * sqrt(5)

the total distance AB = sqrt( 25 + 100) = sqrt(125) = 5 * sqrt(5)

AP/AB = 2*sqrt(5)/[5 * sqrt(5)] = 2/5

yes it checks