Determine the scale factor used in the dilation of line segment AB, centered at the origin.
A(4,-12), B(8,-16)
a)-4
b)1/4
c)4
d)2
To find the scale factor, we can use the distance formula:
The distance between points A(4, -12) and B(8, -16) is given by:
d = √((8-4)^2 + (-16-(-12))^2)
= √(4^2 + (-4)^2)
= √(16 + 16)
= √32
= 4√2
The original length of line segment AB is 4√2.
Now let's find the distance between the origin (0, 0) and the image of point B after dilation:
The image of point B after dilation has coordinates (8k, -16k), where k is the scale factor.
So, the distance between (0, 0) and (8k, -16k) is given by:
d' = √((8k)^2 + (-16k)^2)
= √(64k^2 + 256k^2)
= √(320k^2)
= 4√(5k^2)
= 4k√5
We know that the dilated length of line segment AB is 4k√5.
Therefore, the scale factor used in the dilation of line segment AB is:
Scale factor = (Dilated length) / (Original length)
= (4k√5) / (4√2)
= (k√5) / √2
= k√(5/2)
Since the scale factor is equal to k√(5/2), we can see that k = 1/√(5/2) is the scale factor used in the dilation.
To simplify, we rationalize the denominator:
k = 1/√(5/2) × (√(5/2)/√(5/2))
= √(2/5)
= √2 / √5
= (√2 / √5) × (√5/√5)
= √10 / 5
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is √10 / 5.
None of the provided options match the scale factor √10 / 5.