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
14
2
−4
4
To find the scale factor used in the dilation of line segment AB, we need to compare the lengths of the original line segment AB and the dilated line segment A'B'.
The length of line segment AB can be found using the distance formula:
√((x2 - x1)^2 + (y2 - y1)^2)
= √((2 - 1)^2 + (-4 - (-3))^2)
= √(1^2 + (-1)^2)
= √(1 + 1)
= √2
The length of line segment A'B' can also be found using the distance formula:
√((x2 - x1)^2 + (y2 - y1)^2)
= √((8 - 4)^2 + (-16 - (-12))^2)
= √(4^2 + (-4)^2)
= √(16 + 16)
= √32
= 4√2
The scale factor is then the ratio of the lengths of the dilated line segment to the original line segment:
scale factor = length of A'B' / length of AB
= (4√2) / √2
= 4
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 4.
To determine the scale factor used in the dilation of line segment AB, centered at the origin, we need to compare the lengths of the original line segment AB and the dilated line segment A' B'.
The length of the original line segment AB can be calculated using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, x1 = 1, y1 = -3, x2 = 2, and y2 = -4:
AB = √((2 - 1)^2 + (-4 - (-3))^2)
= √(1^2 + (-4 + 3)^2)
= √(1 + 1^2)
= √(1 + 1)
= √2
The length of the dilated line segment A' B' can be calculated in the same way:
A'B' = √((x2' - x1')^2 + (y2' - y1')^2)
In this case, x1' = 4, y1' = -12, x2' = 8, and y2' = -16:
A'B' = √((8 - 4)^2 + (-16 - (-12))^2)
= √(4^2 + (-16 + 12)^2)
= √(16 + (-4)^2)
= √(16 + 16)
= √32
The scale factor is determined by dividing the length of the dilated segment A'B' by the length of the original segment AB:
Scale factor = A'B' / AB
= √32 / √2
= √(32 / 2)
= √16
= 4
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 4.
To determine the scale factor used in the dilation of line segment AB, centered at the origin, we need to compare the lengths of the original line segment AB and the dilated line segment A' B'.
First, let's find the length of line segment AB using the coordinates provided:
Point A: (1, -3)
Point B: (2, -4)
Using the distance formula, we can calculate the length of AB:
Length AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Length AB = sqrt((2 - 1)^2 + (-4 - (-3))^2)
Length AB = sqrt(1^2 + (-4 + 3)^2)
Length AB = sqrt(1 + 1^2)
Length AB = sqrt(1 + 1)
Length AB = sqrt(2)
Now let's find the length of line segment A' B' using the coordinates provided:
Point A': (4, -12)
Point B': (8, -16)
Using the distance formula, we can calculate the length of A' B':
Length A' B' = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Length A' B' = sqrt((8 - 4)^2 + (-16 - (-12))^2)
Length A' B' = sqrt(4^2 + (-16 + 12)^2)
Length A' B' = sqrt(16 + (-4)^2)
Length A' B' = sqrt(16 + 16)
Length A' B' = sqrt(32)
Length A' B' = sqrt(16 * 2)
Length A' B' = 4 * sqrt(2)
Now, to find the scale factor, we compare the lengths of AB and A' B':
Scale factor = Length A' B' / Length AB
Scale factor = (4 * sqrt(2)) / sqrt(2)
Scale factor = 4
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 4.