Question

A. Given directed line segment MN with M(-6,-2) and N(3,1), find point P that partitions the segment into a ratio of 1:2. Explain your reasoning. B. Given directed line segment JK with J(-1,6) and K(3,-2), find point L that partitions the segment into a ratio of 3:1. Explain your reasoning.

Answer 1

To find the points that partition the line segments into the given ratios, we calculate the coordinates using the differences between the endpoints and multiply them by the ratio. For segment MN with endpoints M(-6,-2) and N(3,1), point P that partitions the segment into a ratio of 1:2 is (-3, -1). For segment JK with endpoints J(-1,6) and K(3,-2), point L that partitions the segment into a ratio of 3:1 is (2, 0).

Step-by-step explanation:

A. Finding Point P

To find the point P that partitions the segment MN into a ratio of 1:2, we need to calculate the coordinates of P. We can do this by finding the difference between the x-coordinates and y-coordinates of M and N and multiplying them by the ratio. Let's calculate:

Difference in x-coordinates: 3 - (-6) = 9

Difference in y-coordinates: 1 - (-2) = 3

Ratio of 1:2 means that the x-coordinate of P is 1/3 of the difference in x-coordinates, and the y-coordinate of P is 1/3 of the difference in y-coordinates. Therefore, the coordinates of P are:

x-coordinate of P: -6 + (1/3 * 9) = -6 + 3 = -3

y-coordinate of P: -2 + (1/3 * 3) = -2 + 1 = -1

So, the point P that partitions the segment MN into a ratio of 1:2 is (-3, -1).

B. Finding Point L

To find the point L that partitions the segment JK into a ratio of 3:1, we can follow the same process. Let's calculate:

Difference in x-coordinates: 3 - (-1) = 4

Difference in y-coordinates: -2 - 6 = -8

Ratio of 3:1 means that the x-coordinate of L is 3/4 of the difference in x-coordinates, and the y-coordinate of L is 3/4 of the difference in y-coordinates. Therefore, the coordinates of L are:

x-coordinate of L: -1 + (3/4 * 4) = -1 + 3 = 2

y-coordinate of L: 6 + (3/4 * -8) = 6 - 6 = 0

So, the point L that partitions the segment JK into a ratio of 3:1 is (2, 0).

Answer 2

1. By definition, a given point that divides a segment into a ratio a:b, as the following x-coordinate:

`x1+(a)/(a+b) (x2-x1)`

And the y-coordinate:

`y1+(a)/(a+b)(y2-y1)`

2. Kepping this on mind, you have:

A) x-coordinate:

`-6+((1)/(1+2)) (3+6)=-3`

y-coordinate:

`-2+((1)/(1+2) )(1+2)=-1`

The answer is:
`P(-3,-1)`

B) The x-coordinate is:

`-1+((3)/(3+1) )(3+1)=2`

The y-coordinate is:

`6+((3)/(3+1) )(-2-6)=0`

The answer is:
`L(2,0)`