Question

line AB has endpoints A (-3, 1) and B (4, 5), find the coordinates of point p on directed line segment AB that partitions AB in the ratio of 5:2

Answer 1

Given

A(-3, 1), B(4, 5)

Find

coordinates of P on AB such that AP/PB = 5/2

Solution

AP/PB = 5/2 . . . . . desired result

2AP = 5PB . . . . . . multiply by 2PB

2(P-A) = 5(B-P) . . . meaning of the above

2P -2A = 5B -5P . . eliminate parentheses

7P = 2A +5B . . . . . collect P terms

P = (2A +5B)/7 . . . .divide by the coefficient of P

P = (2(-3, 1) +5(4, 5))/7 . . . . substitute the given points

P = (-6+20, 2+25)/7 . . . . . . simplify

P = (2, 3 6/7)

Answer 2

Firstly, let's look at this ratio 5:2. Despite what you may think, 5:2 is not equal to 5/2, but rather it is equal to "5 equal parts to 2 equal parts", which make up a total of 7 equal parts. In short, they want us to place point P at 5/7 of line AB.

Firstly, how far is -3 to 4 (the x coordinates)? That would be 7 units. Multiply 5/7 by 7:

$\frac{5}{7}\times \frac{7}{1}=\frac{35}{7}=5$

Next, how far is 1 from 5 (the y-coordinates)? That would be 4 units. Multiply 5/7 by 4:

$\frac{5}{7}\times \frac{4}{1}=\frac{20}{7}=2\frac{6}{7}$

Next, since from -3 to 4 you are increasing, add 5 to -3, which gets you a sum of 2. 2 is the x-coordinate of point P.

Next, since from 1 to 5 you are increasing, add 2 6/7 to 1, which gets you a sum of 3 6/7. 3 6/7 is the y coordinate of point P.

Putting it together, point P is at $(2,3\frac{6}{7})$