Determine the scale factor used in the dilation of line segment AB, centered at the origin.
A: 2,3
B: 4,1
To determine the scale factor used in the dilation of line segment AB, we need to calculate the ratio of the length of the dilated segment to the length of the original segment.
The length of original segment AB can be calculated using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
For point A: (x1, y1) = (2, 3)
For point B: (x2, y2) = (4, 1)
d = √((4 - 2)^2 + (1 - 3)^2)
= √(2^2 + (-2)^2)
= √(4 + 4)
= √8
The length of the dilated segment AB can be calculated in the same way, but using the coordinates of the dilated points:
For point A (dilated): (x1, y1) = (2 * 2, 2 * 3) = (4, 6)
For point B (dilated): (x2, y2) = (2 * 4, 2 * 1) = (8, 2)
dilated_d = √((8 - 4)^2 + (2 - 6)^2)
= √(4^2 + (-4)^2)
= √(16 + 16)
= √32
Now, we can calculate the scale factor as the ratio of the length of the dilated segment to the length of the original segment:
scale factor = dilated_d / d
= √32 / √8
To simplify this, we can multiply both the numerator and denominator by √8:
scale factor = (√32 / √8) * (√8 / √8)
= (√(32 * 8)) / (√(8 * 8))
= (√(256)) / (√(64))
= 16 / 8
= 2
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 2.
It isn't 2, I've tried the answer already.
Apologies for the mistake. Let's calculate the correct scale factor.
The length of original segment AB can be calculated using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
For point A: (x1, y1) = (2, 3)
For point B: (x2, y2) = (4, 1)
d = √((4 - 2)^2 + (1 - 3)^2)
= √(2^2 + (-2)^2)
= √(4 + 4)
= √8
The length of the dilated segment AB can be calculated in the same way, but using the coordinates of the dilated points:
For point A (dilated): (x1, y1) = (2 * 0, 3 * 0) = (0, 0)
For point B (dilated): (x2, y2) = (4 * 0, 1 * 0) = (0, 0)
dilated_d = √((0 - 0)^2 + (0 - 0)^2)
= √(0^2 + 0^2)
= √0
= 0
Now, we can calculate the scale factor as the ratio of the length of the dilated segment to the length of the original segment:
scale factor = dilated_d / d
= 0 / √8
= 0
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 0.