Points A left parenthesis negative 10 comma negative 6 right parenthesis and B left parenthesis 6 comma 2 right parenthesis are the endpoints of ModifyingAbove AB with bar. What are the coordinates of point C on ModifyingAbove AB with bar such that AC is three-fourths the length of ModifyingAbove AB with bar?
A graph is shown with x axis labeled negative 10 to 10 with increments of 1 and y axis labeled 10 to negative 10 with increments of 1. Segment A B is on a set of coordinate axes. A has coordinates left parenthesis negative 10 comma negative 6 right parenthesis. B has coordinates left parenthesis 6 comma 2 right parenthesis.
(1 point)
Responses
left-parenthesis 0 comma negative 1 right-parenthesis
Image with alt text: left-parenthesis 0 comma negative 1 right-parenthesis
left-parenthesis 2 comma 0 right-parenthesis
Image with alt text: left-parenthesis 2 comma 0 right-parenthesis
left-parenthesis negative 2 comma negative 2 right-parenthesis
Image with alt text: left-parenthesis negative 2 comma negative 2 right-parenthesis
left-parenthesis 4 comma 1 right-parenthesis
Image with alt text: left-parenthesis 4 comma 1 right-parenthesis
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Based on the information given, the coordinates of point C on AB such that AC is three-fourths the length of AB would be (2, 0).
To find the coordinates of point C such that AC is three-fourths the length of AB, we can use the midpoint formula. The midpoint of AB can be found by taking the average of the x-coordinates and the average of the y-coordinates of points A and B:
Midpoint coordinates = ((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the coordinates of A are (-10, -6) and the coordinates of B are (6, 2). Substituting these values into the formula, we get:
Midpoint coordinates = ((-10 + 6) / 2, (-6 + 2) / 2)
= (-4 / 2, -4 / 2)
= (-2, -2)
So, the midpoint of AB is (-2, -2).
Now, to find the coordinates of C, we can use the fact that AC is three-fourths the length of AB. Since C is three-fourths of the way from A to the midpoint, we can find the x-coordinate of C by taking three-fourths of the difference between the x-coordinate of the midpoint and the x-coordinate of A, and adding it to the x-coordinate of A. Similarly, we can find the y-coordinate of C by taking three-fourths of the difference between the y-coordinate of the midpoint and the y-coordinate of A, and adding it to the y-coordinate of A.
Let's calculate the coordinates of C:
x-coordinate of C = (3/4) * (x-coordinate of midpoint - x-coordinate of A) + x-coordinate of A
= (3/4) * (-2 - (-10)) + (-10)
= (3/4) * (8) - 10
= 6 - 10
= -4
y-coordinate of C = (3/4) * (y-coordinate of midpoint - y-coordinate of A) + y-coordinate of A
= (3/4) * (-2 - (-6)) + (-6)
= (3/4) * (4) - 6
= 3 - 6
= -3
Therefore, the coordinates of point C are (-4, -3).