1)Let AB be the directed line segment beginning at point A(1 , 1) and ending at point B(-4 , 6). Find the point P on the line segment that partitions the line segment into the segments AP and PB at a ratio of 5:1.

A)(-3 1/6,5 1/6)
B)(1/6,1 5/6) <----
C)(-3 4/5,31)
D)(-2 1/6, 6 1/6)

I think it's B but I'm not completely sure.

B) divides AB in the ratio 1:5, but not 5:1.

look for a choice which is 5/6 from point B instead.

A(1, 1). P(x, y), B(-4, 6).

5u + u = 1. u = 1/6, 5u = 5/6.
x-1 = 5/6(-4-1)
x = -19/6 = -3 1/6.

y-1 = 5/6(6-1)
y = 31/6 = 5 1/6.

Let AB be the directed line segment beginning at point A(5 , 1) and ending at point B(11 , 16). Find the point P on the line segment that partitions the line segment into the segments AP and PB at a ratio of 4:5.

To find the point P that divides the line segment AB into the segments AP and PB at a ratio of 5:1, we can use the concept of section formula.

The section formula states that if a line segment between two points P1(x1, y1) and P2(x2, y2) is divided by a point P(x, y) at a ratio of m:n, then the coordinates of point P can be found using the following formula:

P(x, y) = ( (m*x2 + n*x1) / (m + n), (m*y2 + n*y1) / (m + n) )

In this case, we have point A(1, 1) and point B(-4, 6), and we want to divide the line segment AB at a ratio of 5:1.

Using the section formula, we can compute the coordinates of point P as follows:

P(x, y) = ( (5*(-4) + 1*(1)) / (5 + 1), (5*(6) + 1*(1)) / (5 + 1) )
= ( (-20 + 1) / 6, (30 + 1) / 6 )
= ( -19 / 6, 31 / 6 )

So, the coordinates of point P are (-19/6, 31/6).

Thus, the correct answer is option B)(1/6, 1 5/6).