Final answer:
The scale factor of the dilation from triangle ABC to A'B'C' is calculated by comparing the distances from point A and its dilated point A' to the origin, resulting in a scale factor of 1/3.
Step-by-step explanation:
To determine the scale factor of the dilated triangle A'B'C', we need to compare the distances between corresponding points in the original triangle ABC and the dilated triangle A'B'C'. We are given that point A is at (3,6) and the dilated image of point A is at (1,2). To find the scale factor, we can use the coordinates of these points.
Let's calculate the distance between point A and the origin (0,0) for both the original and dilated triangles:
The distance from A to the origin in the original triangle is given by the formula:
√((x2 - x1)^2 + (y2 - y1)^2) = √((3 - 0)^2 + (6 - 0)^2) = √(9 + 36) = √45
The distance from A' to the origin in the dilated triangle is:
√((x2 - x1)^2 + (y2 - y1)^2) = √((1 - 0)^2 + (2 - 0)^2) = √(1 + 4) = √5
Now that we have both distances, we can find the scale factor by dividing the distance in the dilated triangle by the distance in the original triangle:
Scale factor = Distance A' to origin / Distance A to origin = √5 / √45
To simplify, we take the ratio of the square roots:
Scale factor = √(5/45) = √(1/9) = 1/3
Thus, the scale factor for the dilation of triangle ABC to A'B'C' is 1/3.