Determine the scale factor used in the dilation of line segment AB , centered at the origin.
A-(2,3) B-(4,1)
The scale factor is [ ]
To find the scale factor of the dilation, we need to compare the lengths of the original line segment AB and the dilated line segment A'B'.
The length of AB can be found using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the coordinates of A and B:
d_AB = √((4 - 2)^2 + (1 - 3)^2)
= √(2^2 + (-2)^2)
= √(4 + 4)
= √8
= 2√2
Now let's find the length of A'B' using the same formula.
Since the dilation is centered at the origin, the coordinates of A' and B' will be the same as A and B, but multiplied by the scale factor.
A' = (2, 3) * k
B' = (4, 1) * k
where k is the scale factor.
Let's find the length of A'B':
d_A'B' = √(((4k) - (2k))^2 + ((1k) - (3k))^2)
= √((2k)^2 + (-2k)^2)
= √(4k^2 + 4k^2)
= √(8k^2)
= 2√2k
Since A'B' is the dilated version of AB, the scale factor can be found by taking the ratio:
scale factor = d_A'B' / d_AB = (2√2k) / (2√2) = k.
Therefore, the scale factor is k.