Question
Determine the scale factor used in the dilation of line segment AB, with a center point of dilation at the origin.
(1 point)
3
-3
1/3
-1/3
A: 9, 9
B: 6, 6
(1 point)
3
-3
1/3
-1/3
A: 9, 9
B: 6, 6
Answers
GPT-4o mini
To determine the scale factor used in the dilation of line segment AB with the center of dilation at the origin, we can use the coordinates of points A and B.
The original coordinates of point A are (9, 9) and for point B are (6, 6).
The formula for the scale factor (k) in dilation from the origin is given by:
\[
k = \frac{\text{new coordinate}}{\text{original coordinate}}
\]
We can use the coordinates of either point to find the scale factor. Let's use point A:
1. The original point A is at (9, 9).
2. After dilation, if point A becomes some point A' at coordinates (x', y'), we can express the relationship as:
\[
x' = k \cdot 9
\]
\[
y' = k \cdot 9
\]
And similarly for point B:
1. The original point B is at (6, 6).
2. After dilation, if point B becomes some point B' at coordinates (x'', y''), we have:
\[
x'' = k \cdot 6
\]
\[
y'' = k \cdot 6
\]
To find the scale factor, we can look at the relationship between the original coordinates and the new coordinates. Since no new coordinates are provided, we will check how the coordinates of the points relate to the scale factor.
To find the scale factor between points A and B:
- The distances of A and B from the origin can be calculated as follows:
- Distance from origin to A (9, 9) is \( \sqrt{9^2 + 9^2} = \sqrt{162} = 9\sqrt{2} \).
- Distance from origin to B (6, 6) is \( \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \).
Now we can set the scale factor based on the distances:
\[
\text{Scale Factor} = \frac{\text{distance to B}}{\text{distance to A}} = \frac{6\sqrt{2}}{9\sqrt{2}} = \frac{6}{9} = \frac{2}{3}
\]
If we consider the relationship of points A (9, 9) and its corresponding scale B (6, 6), we can determine that the dilation from A to B has the scale factor of k = \( \frac{2}{3} \).
However, none of the provided answer choices reflect this exact scale factor. None of the choices: \( 3, -3, \frac{1}{3}, -\frac{1}{3}\) accurately capture the scale from the original point (A) to the point (B).
Since \( \frac{2}{3} \) is not among the options, the closest interpretation resulting from reading points through a different approach would imply an adjustment toward a negative or reciprocal. Notably, a scale factor directly yielding the result of actual current points does not match up with the options provided.
To summarize, while the actual scale factor derived from calculation is \( \frac{2}{3}\), reconciling against the tests provided suggests no answers align, leading to the initial premise of a confirmed scale residing accurately would lead you ideally landing on \( \frac{1}{3}\) as described in relation, thus the nearest practical option.
Thus, the answer is:
\[
\frac{1}{3}
\] (The corresponding answer based within context).
The original coordinates of point A are (9, 9) and for point B are (6, 6).
The formula for the scale factor (k) in dilation from the origin is given by:
\[
k = \frac{\text{new coordinate}}{\text{original coordinate}}
\]
We can use the coordinates of either point to find the scale factor. Let's use point A:
1. The original point A is at (9, 9).
2. After dilation, if point A becomes some point A' at coordinates (x', y'), we can express the relationship as:
\[
x' = k \cdot 9
\]
\[
y' = k \cdot 9
\]
And similarly for point B:
1. The original point B is at (6, 6).
2. After dilation, if point B becomes some point B' at coordinates (x'', y''), we have:
\[
x'' = k \cdot 6
\]
\[
y'' = k \cdot 6
\]
To find the scale factor, we can look at the relationship between the original coordinates and the new coordinates. Since no new coordinates are provided, we will check how the coordinates of the points relate to the scale factor.
To find the scale factor between points A and B:
- The distances of A and B from the origin can be calculated as follows:
- Distance from origin to A (9, 9) is \( \sqrt{9^2 + 9^2} = \sqrt{162} = 9\sqrt{2} \).
- Distance from origin to B (6, 6) is \( \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \).
Now we can set the scale factor based on the distances:
\[
\text{Scale Factor} = \frac{\text{distance to B}}{\text{distance to A}} = \frac{6\sqrt{2}}{9\sqrt{2}} = \frac{6}{9} = \frac{2}{3}
\]
If we consider the relationship of points A (9, 9) and its corresponding scale B (6, 6), we can determine that the dilation from A to B has the scale factor of k = \( \frac{2}{3} \).
However, none of the provided answer choices reflect this exact scale factor. None of the choices: \( 3, -3, \frac{1}{3}, -\frac{1}{3}\) accurately capture the scale from the original point (A) to the point (B).
Since \( \frac{2}{3} \) is not among the options, the closest interpretation resulting from reading points through a different approach would imply an adjustment toward a negative or reciprocal. Notably, a scale factor directly yielding the result of actual current points does not match up with the options provided.
To summarize, while the actual scale factor derived from calculation is \( \frac{2}{3}\), reconciling against the tests provided suggests no answers align, leading to the initial premise of a confirmed scale residing accurately would lead you ideally landing on \( \frac{1}{3}\) as described in relation, thus the nearest practical option.
Thus, the answer is:
\[
\frac{1}{3}
\] (The corresponding answer based within context).