1) Segment CD has endpoints (-4, 3) and (8, -1). Find the coordinates of the point that divides the line segment directed from C to D in the ratio of 2:3.
A) (6/5,26/5)
B) (4/5,7/5)
C) (1/5,13/5)
D) (16/5,3/5)
I would provide what I think the answer is- but I'm really stuck and I don't have any idea. The answer is fine, since that's what I need, but I would love some step to step help.
dividing the line into pieces of the ratio 2:3 means that the desired point is 2/5 of the way from C to D, right?
The directed x-distance from C to D is 12
The directed y-distance from C to D is -4
So for the x-value add 2/5 of 12 to C
Same for the y. add 3/5 of -4 to C
I find that most students find the following method very easy to understand for these type of problems:
Let the point be P(x,y)
For the x,
(x+4)/(8-x) = 2/3
3x + 12 = 16 -2x
5x = 4
x = 4/5 <----- I see that as a choice
Now repeat the same steps for y
To find the coordinates of the point that divides the line segment CD in the ratio of 2:3, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two endpoints (x1, y1) and (x2, y2) can be found by taking the average of the x-coordinates and the average of the y-coordinates.
Step 1: Find the x-coordinate of the midpoint.
To find the x-coordinate of the midpoint, we can use the formula:
x = (2*x2 + 3*x1)/(2+3)
Substituting the given coordinates, we have:
x = (2*8 + 3*(-4))/(2+3)
x = (16 - 12)/5
x = 4/5
Step 2: Find the y-coordinate of the midpoint.
To find the y-coordinate of the midpoint, we can use the formula:
y = (2*y2 + 3*y1)/(2+3)
Substituting the given coordinates, we have:
y = (2*(-1) + 3*3)/(2+3)
y = (-2 + 9)/5
y = 7/5
Therefore, the coordinates of the point that divides the line segment CD in the ratio of 2:3 are (4/5, 7/5).
So the answer is option B) (4/5,7/5).