Line segment AB has endpoints A (6, 9) and B (12, 3) And the end coordinates after the dilation are A' (2,3) and B (4,1).Find the dilation with a scale factor centered at the origin.
To find the dilation with a scale factor centered at the origin, we need to find the ratio of the corresponding side lengths in the dilated and original figures.
The length of segment AB is the distance between points A and B. Using the distance formula, we have:
AB = sqrt((12 - 6)^2 + (3 - 9)^2)
= sqrt(6^2 + (-6)^2)
= sqrt(36 + 36)
= sqrt(72)
= 6√2
Similarly, the length of segment A'B' is the distance between points A' and B', which is also given as:
A'B' = sqrt((4 - 2)^2 + (1 - 3)^2)
= sqrt(2^2 + (-2)^2)
= sqrt(4 + 4)
= sqrt(8)
= 2√2
The scale factor is the ratio of the lengths of the segments in the dilated and original figures:
scale factor = A'B' / AB
= (2√2) / (6√2)
= 2/6
= 1/3
Therefore, the dilation with a scale factor centered at the origin is 1/3.
not correct
I apologize for the mistake. Let's find the correct dilation with a scale factor centered at the origin.
To find the scale factor, we can use the formula:
scale factor = distance(A', O) / distance(A, O),
where A' and A are the endpoints of the dilated and original line segment, respectively, and O is the origin (0, 0).
First, let's find the distance from A' to O:
distance(A', O) = sqrt((2 - 0)^2 + (3 - 0)^2)
= sqrt(4 + 9)
= sqrt(13).
Next, let's find the distance from A to O:
distance(A, O) = sqrt((6 - 0)^2 + (9 - 0)^2)
= sqrt(36 + 81)
= sqrt(117).
Now, let's calculate the scale factor:
scale factor = distance(A', O) / distance(A, O)
= sqrt(13) / sqrt(117)
= sqrt(13/117).
Simplifying the square root fraction gives:
scale factor = sqrt(1/9)
= 1/3.
Therefore, the dilation with a scale factor centered at the origin is 1/3.
thats the same answer which is incorrect
I apologize for the confusion. Let's find the correct dilation with a scale factor centered at the origin.
To find the scale factor, we can compare the distances between the corresponding points on the original and dilated line segments.
The distance between points A and B is:
AB = sqrt((12 - 6)^2 + (3 - 9)^2)
= sqrt(36 + 36)
= sqrt(72)
= 6√2.
The distance between points A' and B' is:
A'B' = sqrt((4 - 2)^2 + (1 - 3)^2)
= sqrt(2^2 + (-2)^2)
= sqrt(4 + 4)
= sqrt(8)
= 2√2.
To find the scale factor, we divide the length of the dilated line segment (A'B') by the length of the original line segment (AB):
scale factor = A'B' / AB
= (2√2) / (6√2)
= 2/6
= 1/3.
Therefore, the dilation with a scale factor centered at the origin is 1/3.