Question
Transformations and Similarity Unit Test
1 of 151 of 15 Items
Question
Use the image to answer the question.
A coordinate plane with four quadrants showsthe x-axis ranging from negative 4 to 12 in increments of 1 and the y-axis from 1 to negative 16 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 4. The plotted points are as follows: Upper A is left parenthesis 1 comma negative 3 right parenthesis and upper B is left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 4 comma negative 12 right parenthesis and upper B prime as left parenthesis 8 comma negative 16 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.
(1 point)
Responses
4
4
14
Start Fraction 1 over 4 End Fraction
2
2
−4
1 of 151 of 15 Items
Question
Use the image to answer the question.
A coordinate plane with four quadrants showsthe x-axis ranging from negative 4 to 12 in increments of 1 and the y-axis from 1 to negative 16 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 4. The plotted points are as follows: Upper A is left parenthesis 1 comma negative 3 right parenthesis and upper B is left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 4 comma negative 12 right parenthesis and upper B prime as left parenthesis 8 comma negative 16 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.
(1 point)
Responses
4
4
14
Start Fraction 1 over 4 End Fraction
2
2
−4
Answers
GPT-4o mini
To determine the scale factor used in the dilation of line segment \( AB \), we need to compare the distances between the original points \( A \) and \( B \) and their dilated counterparts \( A' \) and \( B' \).
**Original points:**
- \( A(1, -3) \)
- \( B(2, -4) \)
**Dilated points:**
- \( A'(4, -12) \)
- \( B'(8, -16) \)
**Step 1: Calculate the distance of segment AB.**
The distance \( d_{AB} \) between points \( A \) and \( B \) is given by the distance formula:
\[
d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of points \( A \) and \( B \):
\[
d_{AB} = \sqrt{(2 - 1)^2 + (-4 - (-3))^2} = \sqrt{(1)^2 + (-4 + 3)^2} = \sqrt{1 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
**Step 2: Calculate the distance of segment A'B'.**
Using the same distance formula for points \( A' \) and \( B' \):
\[
d_{A'B'} = \sqrt{(8 - 4)^2 + (-16 - (-12))^2}
\]
Substituting the coordinates of points \( A' \) and \( B' \):
\[
d_{A'B'} = \sqrt{(8 - 4)^2 + (-16 + 12)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
**Step 3: Find the scale factor.**
The scale factor \( k \) is the ratio of the lengths of the dilated segment to the original segment:
\[
k = \frac{d_{A'B'}}{d_{AB}} = \frac{4\sqrt{2}}{\sqrt{2}} = 4
\]
Therefore, the scale factor used in the dilation of line segment \( AB \), centered at the origin, is \( \textbf{4} \).
**Final answer: 4**
**Original points:**
- \( A(1, -3) \)
- \( B(2, -4) \)
**Dilated points:**
- \( A'(4, -12) \)
- \( B'(8, -16) \)
**Step 1: Calculate the distance of segment AB.**
The distance \( d_{AB} \) between points \( A \) and \( B \) is given by the distance formula:
\[
d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of points \( A \) and \( B \):
\[
d_{AB} = \sqrt{(2 - 1)^2 + (-4 - (-3))^2} = \sqrt{(1)^2 + (-4 + 3)^2} = \sqrt{1 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
**Step 2: Calculate the distance of segment A'B'.**
Using the same distance formula for points \( A' \) and \( B' \):
\[
d_{A'B'} = \sqrt{(8 - 4)^2 + (-16 - (-12))^2}
\]
Substituting the coordinates of points \( A' \) and \( B' \):
\[
d_{A'B'} = \sqrt{(8 - 4)^2 + (-16 + 12)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
**Step 3: Find the scale factor.**
The scale factor \( k \) is the ratio of the lengths of the dilated segment to the original segment:
\[
k = \frac{d_{A'B'}}{d_{AB}} = \frac{4\sqrt{2}}{\sqrt{2}} = 4
\]
Therefore, the scale factor used in the dilation of line segment \( AB \), centered at the origin, is \( \textbf{4} \).
**Final answer: 4**