Question

For the directed line segment whose endpoints are (0,0) and (4,3), find the coordinates of the point that partitions the segment into a ratio of 3 to 2. M

Answer 1

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).