determine the scale factor used in the dilation of line segment A 2,3 B 4,1 centured at the origin
To determine the scale factor used in the dilation of line segment AB centered at the origin, we can compare the lengths of the original line segment AB and the dilated line segment.
The length of the original line segment AB can be calculated using the distance formula:
AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
AB = √[(4 - 2)² + (1 - 3)²]
AB = √[(2)² + (-2)²]
AB = √[4 + 4]
AB = √8
AB = 2√2
The length of the dilated line segment can be calculated in the same way. Assuming the scale factor is denoted by "k":
AB' = √[(kx₂ - kx₁)² + (ky₂ - ky₁)²]
AB' = √[(k(4) - k(2))² + (k(1) - k(3))²]
AB' = √[(4k - 2k)² + (k - 3k)²]
AB' = √[(2k)² + (-2k)²]
AB' = √[4k² + 4k²]
AB' = √[8k²]
AB' = √(8) * √(k²)
AB' = 2√2 * k
Comparing the two lengths, we have:
2√2 * k = 2√2
Canceling out the 2√2 terms on both sides, we are left with:
k = 1
Therefore, the scale factor used in the dilation of line segment AB centered at the origin is 1.