Question

Line segment AB has endpoints A(10, 4) and B(2, 8). Find the coordinates of the point that divides the line segment directed from A to B in the ratio of 1:4.
A) (6, 6)
B) (2, 56/5)
C) (24/5, 42/5)
D) (42/5, 24/5)

Answer 1

So despite what you may think, 1:4 is not 1/4 in this situation, but rather it's "1 equal part to 4 equal parts", which is a total of 5 equal parts. In short, they are asking you for the point at the 1/5 mark from A to B.

Firstly, how many units apart is 10 to 2 (the x-coordinates)? That would be 8 units. Multiply 1/5 by 8:

$\frac{1}{5}\times \frac{8}{1}=\frac{8}{5}$

Next, how many units apart is 4 to 8 (the y-coordinates)? That would be 4 units. Multiply 1/5 by 4:

$\frac{1}{5}\times \frac{4}{1}=\frac{4}{5}$

Now, since from 10 to 2 you are decreasing, subtract 8/5 from 10:

$\frac{10}{1}\times \frac{5}{5}=\frac{50}{5}\\\\\frac{50}{5}-\frac{8}{5}=\frac{42}{5}$

42/5 is the x-coordinate.

Now, since from 4 to 8 you are increasing, add 4/5 to 4:

$4+\frac{4}{5}=4\frac{4}{5}=\frac{24}{5}$

24/5 is the y-coordinate.

Putting it together, the point is (42/5, 24/5), or D.