1. A, B and C are collinear, and B is between A and C. The ratio of AB to AC is 3:4. If A is at (-8,1) and B is at (-2,-2), what are the coordinates of C?
2. The segment joining (1,-3) and (4,-6) is extended a distance equal to one-sixth of its own length. Find the terminal point.
I solved these two problems yet neither of them is correct.
(0,-3)
I checked, its (0, -3)
Let's reevaluate the solutions to the problems:
1. Given that A, B, and C are collinear and B is between A and C, we can use the concept of proportionality to find the coordinates of point C. We know that the ratio of AB to AC is 3:4, meaning that the length of AB is 3 units and the length of AC is 4 units.
To find the coordinates of C, we need to consider the change in the x-coordinate and the change in the y-coordinate from A to B. We can then use these changes to determine the coordinates of C.
The change in the x-coordinate from A to B is: Δx = -2 - (-8) = 6
The change in the y-coordinate from A to B is: Δy = -2 - 1 = -3
To find the change in x-coordinate and y-coordinate from A to C, we need to scale the changes by the ratio 4:3 since AC is 4 units and AB is 3 units.
The change in the x-coordinate from A to C is: (4/3) * Δx = (4/3) * 6 = 8
The change in the y-coordinate from A to C is: (4/3) * Δy = (4/3) * (-3) = -4
Now, we can find the coordinates of C by adding these changes to the coordinates of point A:
x-coordinate of C: -8 + 8 = 0
y-coordinate of C: 1 + (-4) = -3
Therefore, the coordinates of point C are (0, -3).
2. To find the terminal point when the segment joining (1,-3) and (4,-6) is extended a distance equal to one-sixth of its own length, we can use the concept of a linear equation.
The equation of the line passing through the two given points can be found using the slope-intercept form:
y - y1 = m(x - x1)
Given the points (1,-3) and (4,-6):
m = (y2 - y1) / (x2 - x1) = (-6 - (-3)) / (4 - 1) = -3/3 = -1
Using the point-slope form, we have:
y - (-3) = -1(x - 1)
y + 3 = -x + 1
y = -x - 2
To extend the segment a distance equal to one-sixth of its own length, we need to find the length of the segment and then add one-sixth of that length to the coordinates of the endpoint.
The length of the segment AB is given by the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((4 - 1)^2 + (-6 - (-3))^2)
d = sqrt(9 + 9)
d = sqrt(18)
To find one-sixth of the length, we divide d by 6:
one-sixth of the length = sqrt(18) / 6 = sqrt(3)
Now, we add this one-sixth of the length to the coordinates of point B:
x-coordinate of the terminal point = 4 + sqrt(3)
y-coordinate of the terminal point = -6 + sqrt(3)
Therefore, the coordinates of the terminal point are (4 + sqrt(3), -6 + sqrt(3)).
If the solutions provided previously were incorrect, I apologize for any confusion caused. Please double-check your calculations and steps to ensure accuracy.
AB:AC = 3:4 means that B is 3/4 of the way from A to C.
Think about it. Start at A. When you get to B, AB is 3/4 of the whole distance AC.
B = A + k (C-A)
when k=0, you are at A
when k=1, you are at C
Bx = Ax + 3/4 (Cx - Ax) = -8 + 3/4 (-2+8) = -8 + 3/4 (6) = -7/2
That is, the distance from -8 to -2 is 6. 3/4 of 6 = 9/2
By = 1 + 3/4 (-2-1) = 1 + 3/4 (-3) = -5/4
For #2, If we let A = (1,-3) and B = (4,-6), we want C such that BC = 1/6 AB
That is, AC:AB = 7/6
Cx = Ax + 7/6 (Bx - Ax) = 1 + 7/6 (4-1) = 9/2
Cy = -3 + 7/6 (-6+3) = -3 + 7/6 (-3) = -13/2
So, C = (9/2, -13/2)
You really should get out some graph paper and plot the points, to make sure your answer satisfies the conditions. It may be easier to see geometrically.