Line AB has endpoints of A(-2, -5) and B(6, 11). Point P is on segment AB such that AP : PB = 3 : 1. Determine and state the coordinates of point P.

Answer 1

If the point partitions the line internally then the formula used is: ( (m x2 + n x1) / (m+n) , (m y2 + n y1) / (m+n) ), If we consider m : n is the ratio and point A as (x1 , y1) and point B as (x2, y2).

If the point partitions the line externally then the formula used is: ( (m x2 - n x1) / (m-n) , (m y2 - n y1) / (m-n) ), If we consider m : n is the ratio and point A as (x1 , y1) and point B as (x2, y2).

Coming to the question, A B as end -points and P is on the segment , AP : PB is given and we have to find point P.

So, P partitions the line AB internally.

P(x3 , y3) = ( (m x2 + n x1) / (m+n) , (m y2 + n y1) / (m+n) ).

Solve the problem, if you have any further doubts, you can message me, i am available anytime to clarify.

Answer 2

Since the ratio between line segments AP ad PB is 3:1, point AP is 3/4 the distance from point A. You can use the coordinates of A and B to find the slope of line segment AB, which is the change of y divided by the change of x. Those x and y values also form two side of a right triangle with AB as the hypotenuse. Using the 3/4 ratio for the distance from point A to point P, we can also construct another right triangle that is proportional to the first one. Then by applying the principle of proportionality we can determine the lengths of the sides of the new triangle because all the sides will have the same proportions, and from that we can find the coordinate points of point P.

Here's how:

The horizontal side of the first triangle is the change of x between point A and B, That would be 6 - (_2) =8. The x coordinate of point P would be a change 3/4 of that from point A, 8 x 3/4 =6. So the x coordinate is -2 + 6 =4.

Now do the same for Y. The change of y from A to B is 11 - (-5) = 16. Take 3/4 of that to find the change of y from point A to point P, so 16 x 3/4 = 12. Now use 12 to find the y coordinate, starting from point A, that gives -5 + 12 = 7

So the coordinate of point P are (4,7).

Hope this helps