Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(−6, −9), B(2, −5)

Question

The ratio is 1to 3

The coordinates are (   ,   )

Answer 1

Imagine a right triangle with point A as the lower left vertex and B as the upper right vertex. The hypotenuse is the line from A to B. Points A, P and B lie on the hypotenuse and point O is the 90 degree vertex. Note that the point directly below P forms another vertex and it is named OO. It is given that the distance AP divided by the distance PB = 3/2. Note that AP+PB = AB thus AP=3/5AB and PB= 2/4AB. Note that A forms the lower left vertex for two triangles BAO and PAOO. By similar triangles, AOO = 3/5AO = 3/5(14-4)=6 and BOO=2/5BO= 2/5(15-10)=2. Thus the coorinates of P are (4+6,10+2), P(10,12).