Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(4, 10), B(14, 15); 3 to 2 The coordinates of P are ( , ).

Answer 1

HI Emiliana Q

P(10, 13)

Since the point P is after point A and before point B

A____________________P____________B

AP = P - A

PB = B - P

We need find the coordinates of P

Let AP_x = x_{p -} x_a = x - 4

Let AP_y = y_p - y_a = y - 10

Let P_xB = x_b - x_p = 14 - x

Let P_yB = y_b - y_p = 15 - y

Now we just set the ratio of the coordinates equal to 3/2 and cross multiply to find the coordinates of P

For AP_x/P_xB = 3/2

(x - 4)/(14 - x) = 3/2

2x - 8 = 42 - 3x

2x + 3x = 42 + 8

5x = 50

x = 10 = P_x

For AP_y/P_yB = 3/2

(y -10)/(15 - y) = 3/2

2y - 20 = 45 - 3y

2y + 3y = 45 + 20

5y = 65

y = 13 = P_y

Now check the ratio

AP_x/P_xB = (10 - 4)/(14 - 10)= 6/4 = 3/2

AP_y/P_yB = (13 -10)/(15 - 13) = 3/2

Answer 2

P is (x, y). So AP/PB = 3/2; (x - 4)/(14 - x) = 3/2; 3(14 - x) = 2(x - 4); 42 - 3x = 2x - 8; 50 = 5x , x = 10

(y - 10)/(15 - y) = 3/2, 2(y - 10) = 3(15 - y), 2y - 20 = 45 - 3y; 5y = 65, y = 13.

P = (10, 13).