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

To find the coordinates of the point that partitions the line segment into a ratio of 3 to 2, use the section formula.

Step-by-step explanation:

To find the coordinates of the point that partitions the line segment into a ratio of 3 to 2, we can use the section formula. The coordinates of the point M can be found using the formula:

x = (x1 * k + x2 * m) / (k + m)

y = (y1 * k + y2 * m) / (k + m)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints, and k and m are the ratios. In this case, the coordinates of the endpoints are (0,0) and (4,3), and the ratio is 3:2.

Plugging the values into the formula, we get:

x = (0 * 3 + 4 * 2) / (3 + 2) = 8/5 = 1.6

y = (0 * 3 + 3 * 2) / (3 + 2) = 6/5 = 1.2

So, the coordinates of the point that partitions the line segment into a ratio of 3 to 2 are (1.6, 1.2).

Answer 2

(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).