Question

Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(10, 8), B(18, 12); 1 to 3

Answer 1

Answer:

The coordinates of P are (12,9)

Step-by-step explanation:

Equations

We are given the endpoints of the segment AB: A(10,8) B(18,12).

It's required to find the coordinates of point P(x,y) along the segment AB such that:

$\displaystyle \frac{AP}{PB}=\frac{1}{3}$

The ratio of the distances is the same as the ratio of their respective coordinates:

$\displaystyle \frac{x_{AP}}{x_{PB}}=\frac{1}{3}$

$\displaystyle \frac{y_{AP}}{y_{PB}}=\frac{1}{3}$

Since

$x_{AP}=x_P-x_A=x-10$

$x_{PB}=x_B-x_P=18-x$

Then:

$\displaystyle \frac{x-10}{18-x}=\frac{1}{3}$

Multiplying by 3 (18-x):

3( x - 10 )= 18 - x

3x - 30 = 18 - x

Adding x and 30:

4x = 48

x = 12

Similarly:

$\displaystyle \frac{y-8}{12-y}=\frac{1}{3}$

Multiplying by 3 (12-y):

3( y - 8 ) = 12 - y

3y - 24 = 12 - y

Adding y and 24:

4y = 36

y = 9

The coordinates of P are (12,9)