Determine the scale factor used in the dilation of line segment AB, centered at the origin.
A.2
B. -1/4
C. 4
D. -4
To determine the scale factor used in the dilation, we can look at the coordinates of the original line segment AB and compare it to the coordinates of the dilated line segment.
If the original line segment AB is centered at the origin, then the coordinates of point A and point B would be (a, b) and (-a, -b) respectively, where a and b are positive.
If the scale factor of the dilation is k, then the coordinates of the dilated line segment would be (ka, kb) and (-ka, -kb).
Comparing the coordinates, we can see that for the x-coordinate, the original value is a and the dilated value is ka. Thus, the scale factor for the x-coordinate is k.
Likewise, for the y-coordinate, the original value is b and the dilated value is kb. Thus, the scale factor for the y-coordinate is k.
Therefore, the overall scale factor used in the dilation of line segment AB is k.
Since the options for the scale factor are 2, -1/4, 4, and -4, we need to determine the value of k.
Since the original line segment is centered at the origin, the length of the line segment AB is given by the distance formula:
Length of AB = √((2a)^2 + (2b)^2) = √(4a^2 + 4b^2) = 2√(a^2 + b^2)
For the dilated line segment, the length would be:
Length of dilated AB = √((2ka)^2 + (2kb)^2) = √(4k^2a^2 + 4k^2b^2) = 2√(k^2(a^2 + b^2))
Since the original length and the dilated length are proportional, we can write the equation:
2√(a^2 + b^2) = 2√(k^2(a^2 + b^2))
Cancelling out the common factors, we get:
√(a^2 + b^2) = √(k^2(a^2 + b^2))
Squaring both sides to get rid of the square root, we get:
a^2 + b^2 = k^2(a^2 + b^2)
Dividing both sides by (a^2 + b^2), we get:
1 = k^2
Since a and b are both positive, k must also be positive.
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is +1.
None of the given options match with the answer, so we cannot determine the scale factor from the options given.