Question
Determine the scale of factor using the dilation of a line statement, a centered at the origin
Answers
GPT 3.5
To determine the scale factor using the dilation of a line statement centered at the origin, you would compare the lengths of the original line segment and the corresponding dilated line segment.
Let's say the original line segment is AB with coordinates A(x1, y1) and B(x2, y2), and the dilated line segment is A'B' with coordinates A'(kx1, ky1) and B'(kx2, ky2), where k is the scale factor.
To find the scale factor, we would calculate the ratio of the lengths of the original line segment to the dilated line segment:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
A'B' = √[(kx2 - kx1)^2 + (ky2 - ky1)^2]
Now, we can set up the ratio as follows:
AB / A'B' = √[(x2 - x1)^2 + (y2 - y1)^2] / √[(kx2 - kx1)^2 + (ky2 - ky1)^2]
Simplify further by removing the square roots:
AB / A'B' = [(x2 - x1)^2 + (y2 - y1)^2] / [(kx2 - kx1)^2 + (ky2 - ky1)^2]
Since A and B are centered at the origin, x1=x2=y1=y2=0, thus:
AB / A'B' = [0 + 0] / [(k*0 - k*0)^2 + (k*0 - k*0)^2]
AB / A'B' = 0 / 0
The scale factor, k, is undefined because the comparison results in division by zero in this particular scenario.
Let's say the original line segment is AB with coordinates A(x1, y1) and B(x2, y2), and the dilated line segment is A'B' with coordinates A'(kx1, ky1) and B'(kx2, ky2), where k is the scale factor.
To find the scale factor, we would calculate the ratio of the lengths of the original line segment to the dilated line segment:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
A'B' = √[(kx2 - kx1)^2 + (ky2 - ky1)^2]
Now, we can set up the ratio as follows:
AB / A'B' = √[(x2 - x1)^2 + (y2 - y1)^2] / √[(kx2 - kx1)^2 + (ky2 - ky1)^2]
Simplify further by removing the square roots:
AB / A'B' = [(x2 - x1)^2 + (y2 - y1)^2] / [(kx2 - kx1)^2 + (ky2 - ky1)^2]
Since A and B are centered at the origin, x1=x2=y1=y2=0, thus:
AB / A'B' = [0 + 0] / [(k*0 - k*0)^2 + (k*0 - k*0)^2]
AB / A'B' = 0 / 0
The scale factor, k, is undefined because the comparison results in division by zero in this particular scenario.