Question

What are the coordinates of the point on the directed line segment from (-6, -1)(−6,−1) to (6, -9)(6,−9) that partitions the segment into a ratio of 3 to 1?

Answer 1

The coordinates of the point that divides the line segment from (-6, -1) to (6, -9) in the ratio 3:1 are (3, -7) by using the section formula.

Step-by-step explanation:

To find the coordinates of the point that divides the segment from (-6, -1) to (6, -9) in the ratio 3:1, we can use the section formula. We will apply the formula for internal division which is given by:

\[(x, y) = \left(\frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n}\right)\]

Here, m:n is the given ratio, (x1, y1) = (-6, -1) are the coordinates of the first point, and (x2, y2) = (6, -9) are the coordinates of the second point. Substituting the values we get:

\[(x, y) = \left(\frac{3 \cdot 6 + 1 \cdot (-6)}{3 + 1}, \frac{3 \cdot (-9) + 1 \cdot (-1)}{3 + 1}\right)\] \[= \left(\frac{18 - 6}{4}, \frac{-27 - 1}{4}\right)\] \[= \left(\frac{12}{4}, \frac{-28}{4}\right)\] \[= (3, -7)\]

Therefore, the coordinates of the point that divides the segment in the ratio of 3 to 1 are (3, -7).

Answer 2

Given:

A segments is from (−6,−1) to (6, -9).

A point divides the line segment into a ratio of 3 to 1.

To find:

The coordinates of the points.

Solution:

Section formula: If a point divides a line segment in m:n, then

`Point=\left((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n)\right)`

The point divides the line segment in 3:1, so by using section formula, we get

`Point=\left((3(6)+1(-6))/(3+1),(3(-9)+1(-1))/(3+1)\right)`

`Point=\left((18-6)/(4),(-27-1)/(4)\right)`

`Point=\left((12)/(4),(-28)/(4)\right)`

`Point=\left(3,-7\right)`

Therefore, the coordinates of the required point are (3,-7).