Question
A dilation of line AB occurs using a scale factor of 1/2 and a center of dilation at the origin. Prove that line A'B' is half the length of line AB by finding the points A' and B'.
A. A' is at (4,8) and B' is at (12,12)
B. A' is at (1,5) and B' is at (3,3)
C. A' is at (1,2) and B' is at (3,3).
D. A' is at (-1,-2) and B' is at (-3, -3)
A. A' is at (4,8) and B' is at (12,12)
B. A' is at (1,5) and B' is at (3,3)
C. A' is at (1,2) and B' is at (3,3).
D. A' is at (-1,-2) and B' is at (-3, -3)
Answers
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Answered by
GPT-4o mini
To find the images of points A and B after the dilation around the origin with a scale factor of \( \frac{1}{2} \), we will use the following formula for dilation:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the new coordinates after dilation, and \( k \) is the scale factor.
**Let’s check the points provided.**
1. **Option A:** A' is at (4,8) and B' is at (12,12)
If \( A = (4, 8) \) and \( B = (12, 12) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 4, \frac{1}{2} \cdot 8\right) = (2, 4) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 12, \frac{1}{2} \cdot 12\right) = (6, 6) \)
Hence, A' should be (2,4) and B' should be (6,6). So, this option is incorrect.
2. **Option B:** A' is at (1,5) and B' is at (3,3)
If \( A = (1, 5) \) and \( B = (3, 3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 1, \frac{1}{2} \cdot 5\right) = \left(\frac{1}{2}, \frac{5}{2}\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 3, \frac{1}{2} \cdot 3\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \)
Hence, A' = (1/2, 2.5) and B' = (1.5, 1.5). This option is incorrect.
3. **Option C:** A' is at (1,2) and B' is at (3,3)
If \( A = (1, 2) \) and \( B = (3, 3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 1, \frac{1}{2} \cdot 2\right) = \left(\frac{1}{2}, 1\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 3, \frac{1}{2} \cdot 3\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \)
Hence, A' = (1/2, 1) and B' = (1.5, 1.5). This option is incorrect.
4. **Option D:** A' is at (-1,-2) and B' is at (-3, -3)
If \( A = (-1, -2) \) and \( B = (-3, -3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot (-1), \frac{1}{2} \cdot (-2)\right) = \left(-\frac{1}{2}, -1\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot (-3), \frac{1}{2} \cdot (-3)\right) = \left(-\frac{3}{2}, -\frac{3}{2}\right) \)
Hence, A' = (-1/2, -1) and B' = (-1.5, -1.5). This option is incorrect.
None of the options given correspond to the correct dilation of line AB with a scale factor of \( 1/2 \).
Nonetheless, to prove that the length of line A'B' is half the length of line AB, we need the coordinates of A and B originally given, for which no options provide that.
To prove this with existing coordinates:
Let \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \). The distance AB is given by:
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\]
The length of A'B' after dilation of A and B:
\[
A'B' = \sqrt{(kx_B - kx_A)^2 + (ky_B - ky_A)^2} = k \cdot \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \frac{1}{2} \cdot AB.
\]
Thus, the length of A'B' is indeed half the length of AB due to the properties of dilation.
\[
(x', y') = (k \cdot x, k \cdot y)
\]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the new coordinates after dilation, and \( k \) is the scale factor.
**Let’s check the points provided.**
1. **Option A:** A' is at (4,8) and B' is at (12,12)
If \( A = (4, 8) \) and \( B = (12, 12) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 4, \frac{1}{2} \cdot 8\right) = (2, 4) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 12, \frac{1}{2} \cdot 12\right) = (6, 6) \)
Hence, A' should be (2,4) and B' should be (6,6). So, this option is incorrect.
2. **Option B:** A' is at (1,5) and B' is at (3,3)
If \( A = (1, 5) \) and \( B = (3, 3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 1, \frac{1}{2} \cdot 5\right) = \left(\frac{1}{2}, \frac{5}{2}\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 3, \frac{1}{2} \cdot 3\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \)
Hence, A' = (1/2, 2.5) and B' = (1.5, 1.5). This option is incorrect.
3. **Option C:** A' is at (1,2) and B' is at (3,3)
If \( A = (1, 2) \) and \( B = (3, 3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 1, \frac{1}{2} \cdot 2\right) = \left(\frac{1}{2}, 1\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 3, \frac{1}{2} \cdot 3\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \)
Hence, A' = (1/2, 1) and B' = (1.5, 1.5). This option is incorrect.
4. **Option D:** A' is at (-1,-2) and B' is at (-3, -3)
If \( A = (-1, -2) \) and \( B = (-3, -3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot (-1), \frac{1}{2} \cdot (-2)\right) = \left(-\frac{1}{2}, -1\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot (-3), \frac{1}{2} \cdot (-3)\right) = \left(-\frac{3}{2}, -\frac{3}{2}\right) \)
Hence, A' = (-1/2, -1) and B' = (-1.5, -1.5). This option is incorrect.
None of the options given correspond to the correct dilation of line AB with a scale factor of \( 1/2 \).
Nonetheless, to prove that the length of line A'B' is half the length of line AB, we need the coordinates of A and B originally given, for which no options provide that.
To prove this with existing coordinates:
Let \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \). The distance AB is given by:
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\]
The length of A'B' after dilation of A and B:
\[
A'B' = \sqrt{(kx_B - kx_A)^2 + (ky_B - ky_A)^2} = k \cdot \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \frac{1}{2} \cdot AB.
\]
Thus, the length of A'B' is indeed half the length of AB due to the properties of dilation.
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