Use the image to answer the question.
A coordinate plane with four quadrants shows x-axis ranging from negative 5 to 12 in increments of 1 and the y-axis from 17 to negative 4 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 1. The plotted points are as follows: Upper A is left parenthesis 2 comma 3 right parenthesis and upper B is left parenthesis 4 comma 1 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 6 comma 9 right parenthesis and upper B prime as left parenthesis 12 comma 3 right parenthesis. The points are joined by a line.
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 can compare the length of AB to the length of A' B'.
The length of AB can be calculated using the distance formula:
AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(4 - 2)² + (1 - 3)²]
= √[2² + (-2)²]
= √(4 + 4)
= √8
= 2√2
The length of A'B' can be calculated using the distance formula:
A'B' = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(12 - 6)² + (3 - 9)²]
= √[6² + (-6)²]
= √(36 + 36)
= √72
= 6√2
To determine the scale factor, we divide the length of A'B' by the length of AB:
Scale factor = A'B' / AB
= (6√2) / (2√2)
= 6/2
= 3
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 3.