Answer:
(12/5, 9/5)
Step-by-step explanation:
Using the Pythagorean Theorem, we can calculate the distance from (0, 0) to (4, 3) as follows: √[ 3^2 + 4^2 ] = 5.
If a point partitions this directed line segment into a ratio of 3:2, then we measure 3 units along this line, plot a point there, and then continue 2 more units to (4, 3).
Next we must find the coordinates of this point. We have a right triangle of hypotenuse 5 (see the work done above). The smaller triangle, which is similar to the larger triangle. has a hypotenuse of 3 (see the previous paragraph), and the two triangles are similar. Thus, the equation of ratios
3 x
------- = ------- is true, and its solution comes from cross multiplication:
5 4
5x = 12, or x = 12/5.
We find the y coordinate of the point in question in the same way:
3 y
------- = ------- is true, and its solution comes from cross multiplication:
5 3
Then 5y = 9, and the y-coordinate is y = 9/5.
The coordinates of the point in question are (12/5, 9/5).
