On a grid, there are two lines with these points =
Line 1 : A = (1, -3)
B = (2, -4)
Line 2 : A = (4, -12)
B = (8, -16)
Determine the scale factor used in the dilation of line segment AB , centered at the origin.
To determine the scale factor used in the dilation of line segment AB, we need to find the ratio of the lengths of the line segments before and after the dilation.
The length of line segment AB before dilation can be found using the distance formula:
AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
For Line 1, the length of AB before dilation is:
AB = √[(2 - 1)² + (-4 - (-3))²] = √[1² + (-1)²] = √[1 + 1] = √2
For Line 2, the length of AB before dilation is:
AB = √[(8 - 4)² + (-16 - (-12))²] = √[4² + (-4)²] = √[16 + 16] = √32 = 4√2
To find the scale factor, we compare the lengths of AB before and after dilation.
For Line 1, let's say the length of AB after dilation is k times the original length AB. The length after dilation is also equal to the distance between A' and B'.
A' = (k * 1, k * (-3)) = (k, -3k)
B' = (k * 2, k * (-4)) = (2k, -4k)
The length of A'B' can be found using the distance formula:
A'B' = √[(2k - k)² + (-4k - (-3k))²] = √[k² + (-k)²] = √[k² + k²] = √[2k²] = k√2
Comparing this to the original length AB, we have:
k√2 = √2
This implies that k = 1. Therefore, the scale factor used in the dilation of Line 1 is 1.
For Line 2, following the same steps:
A' = (k * 4, k * (-12)) = (4k, -12k)
B' = (k * 8, k * (-16)) = (8k, -16k)
The length of A'B' can be found using the distance formula:
A'B' = √[(8k - 4k)² + (-16k - (-12k))²] = √[16k² + 16k²] = √[32k²] = 4√2 * k
Comparing this to the original length AB, we have:
4√2 * k = 4√2
This implies that k = 1. Therefore, the scale factor used in the dilation of Line 2 is also 1.
In conclusion, the scale factor used in the dilation of line segment AB, centered at the origin, is 1 for both Line 1 and Line 2.
Pick between these answer choices,
A) -4
B) 1/4
C) 4
D) 2