Determine the scale factor used in the dilation of segment AB, with a center point of dilation at the origin.
To determine the scale factor used in the dilation of segment AB, we need to compare the lengths of the original segment AB and the dilated segment.
Let's say the length of the original segment AB is x.
After the dilation, the length of the dilated segment will be kx, where k is the scale factor.
Since the center point of dilation is at the origin, the origin is the fixed point during the dilation. This means that any point on segment AB will move along a straight line that passes through the origin.
Therefore, we can use the distance formula to calculate the lengths of segment AB and the dilated segment, using the coordinates of A and B.
Let the coordinates of point A be (x1, y1) and the coordinates of point B be (x2, y2).
The length of segment AB can be calculated using the distance formula:
Length of AB = √((x2 - x1)^2 + (y2 - y1)^2)
Similarly, the length of the dilated segment can be calculated using the distance formula with the coordinates of the dilated points:
Length of dilated segment = √((kx2 - kx1)^2 + (ky2 - ky1)^2)
To find the scale factor, we can divide the length of the dilated segment by the length of segment AB:
k = (Length of dilated segment) / (Length of AB)
Simplifying the equation further, we can factor out the k from the squared terms:
k = (√(k^2(x2 - x1)^2 + k^2(y2 - y1)^2)) / (√((x2 - x1)^2 + (y2 - y1)^2))
Since k is common in the numerator and the denominator, we can cancel it out:
k = √(k^2(x2 - x1)^2 + k^2(y2 - y1)^2) / √((x2 - x1)^2 + (y2 - y1)^2)
To simplify the expression further, we can remove the square roots by squaring both sides:
k^2 = (k^2(x2 - x1)^2 + k^2(y2 - y1)^2) / ((x2 - x1)^2 + (y2 - y1)^2)
Now, we can cancel out the k^2 from the numerator and the denominator:
k^2 = (x2 - x1)^2 + (y2 - y1)^2 / (x2 - x1)^2 + (y2 - y1)^2
Finally, taking the square root of both sides gives us the scale factor:
k = √((x2 - x1)^2 + (y2 - y1)^2) / √((x2 - x1)^2 + (y2 - y1)^2)
Therefore, the scale factor used in the dilation of segment AB with a center point of dilation at the origin is 1.