Given points A(1,3) and B(8,4) , find the coordinates of point P along the directed line segment AB so that AP to PB is the ratio 4 to 1.

Answer 1

Hi Tyrell T.

coordinates of P are (6.6, 3.8)

You are given points A (1,3) and B (8, 4) along directed line segment AB you want to find the coordinates of point P somewhere between point s A and B such that AP to PB is 4 to 1

AP/PB = 4/1

A_(1,3)________________________P•________(8,4)B

Point A comes before point P

Point B comes after point P

Let the distance from A to P = AP

Let the distance from P to B = PB

so the x coordinate of point P is

AP_x = x - 1

PB_x = 8 - x

(x - 1)/(8 -x ) = 4 /1

Cross multiplying we have

x - 1 = 32 - 4x

Combine like terms

x + 4x = 32 + 1

5x = 33

x = 33/5

x = 6.6 is the x coordinate of point P

The y coordinate of point P is

APy = y - 3

PB_y = 4 - y

(y - 3)/(4 - y) = 4/1

Cross Multiplying

y - 3 = 16 - 4y

Combining like terms

y + 4y = 16 + 3

5y = 19

y = 19/5

y = 3.8

So the coordinates of Point P are ( 6.6, 3.8) now we can check the ratio

The x ratio of AP/PB

(6.6 - 1)/ (8 - 6.6) = 5.6/1.4 = 4

The y ratio of AP/PB

(3.8 -3)/(4 - 3.8) = 0.8/0.2 = 4

There are other ways to do the problem but I do hope this helps

Answer 2

You're trying to find 4/5 of the way from A to B. Instead of finding 4/5 of the way between the two points, break the problem up. Find 4/5 of the way between the x coordinates (A_x and B_x which are 1 and 8) and 4/5 of the way between the y coordinates (A_y and B_y which are 3 and 4). These will give you the x and y coordinates of P