Determine the scale factor used in the dilation of line segment AB, centered at the origin.
A = (1, -3)
B = (2, -4)
(1 point)
responses
4
-4
2
1/4
To determine the scale factor used in the dilation, we need to find the ratio of the length of the image of line segment AB to the length of the original AB.
The length of line segment AB can be found using the distance formula.
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
AB = √[(2 - 1)^2 + (-4 - (-3))^2]
AB = √[1^2 + (-1)^2]
AB = √[1 + 1]
AB = √2
Now, let's find the length of the image of line segment AB. Since the dilation is centered at the origin, the image will be on the same line passing through the origin.
The length of the image of AB can be found by multiplying the length of AB by the scale factor.
Image of AB = Scale factor * AB
Let's find the coordinates of the image of A and B.
Image of A = (1 * Scale factor, -3 * Scale factor) = (Scale factor, -3Scale factor)
Image of B = (2 * Scale factor, -4 * Scale factor) = (2Scale factor, -4Scale factor)
The length of the image of AB can be found using the distance formula.
Image of AB = √[(2Scale factor - Scale factor)^2 + (-4Scale factor - (-3Scale factor))^2]
Image of AB = √[Scale factor^2 + (-1)^2Scale factor^2]
Image of AB = √[(1 + 1)Scale factor^2]
Image of AB = √[2Scale factor^2]
Image of AB = √2Scale factor
Now, let's find the ratio of the length of the image of AB to the length of AB.
Scale factor = (Image of AB) / AB
Scale factor = (√2Scale factor) / (√2)
Scale factor = Scale factor / 1
The scale factor is equal to 1 or 1/1.
Therefore, the correct response is 1/4.
Determine the scale factor used in the dilation of line AB, with a center point of dilation at the origin. (1 point)
Responses
A = (9, 9)
B = (6, 6)
3
-1/3
1/3
−3
To determine the scale factor used in the dilation, we need to find the ratio of the distance between the image of point A and the origin to the distance between point A and the origin.
The distance from the origin to point A can be found using the distance formula:
OA = √[(x1 - 0)^2 + (y1 - 0)^2]
OA = √[(9 - 0)^2 + (9 - 0)^2]
OA = √[9^2 + 9^2]
OA = √[81 + 81]
OA = √[162]
OA = 9√2
Now, let's find the coordinates of the image of point A. Since the dilation is centered at the origin, the image will be on the same line passing through the origin.
Image of A = (Scale factor * x-coordinate of A, Scale factor * y-coordinate of A)
Image of A = (Scale factor * 9, Scale factor * 9)
The distance from the origin to the image of point A can be found using the distance formula:
Image of OA = √[(Scale factor * 9 - 0)^2 + (Scale factor * 9 - 0)^2]
Image of OA = √[(9Scale factor)^2 + (9Scale factor)^2]
Image of OA = √[81Scale factor^2 + 81Scale factor^2]
Image of OA = √[(81 + 81)Scale factor^2]
Image of OA = √[162Scale factor^2]
Image of OA = √162Scale factor
Now, let's find the ratio of the image of OA to OA to determine the scale factor:
Scale factor = (Image of OA) / OA
Scale factor = (√162Scale factor) / (9√2)
Scale factor = Scale factor / (9√2 / √162)
Scale factor = Scale factor / (9 / √2 * √162)
Scale factor = Scale factor / (9 / √(2 * 162))
Scale factor = Scale factor / (9 / √(324))
Scale factor = Scale factor / (9 / 18)
Scale factor = Scale factor / (1/2)
Scale factor = Scale factor * 2
Thus, the scale factor is equal to 2.
Therefore, the correct response is 2.
Determine the scale factor used in the dilation of line AB, with a center point of dilation at the origin. (1 point)
Responses
A = (9, 9)
B = (6, 6)
3
-1/3
1/3
−3
Which answer is it?
Apologies for the previous incorrect response. Let's calculate the scale factor correctly this time.
To find the scale factor, we need to determine how much the distances between the points change after the dilation.
The distance between the origin and point A can be found using the distance formula:
OA = √[(x1 - 0)^2 + (y1 - 0)^2]
OA = √[(9 - 0)^2 + (9 - 0)^2]
OA = √[9^2 + 9^2]
OA = √[81 + 81]
OA = √162
OA = 9√2
The distance between the origin and point B can also be found using the distance formula:
OB = √[(x2 - 0)^2 + (y2 - 0)^2]
OB = √[(6 - 0)^2 + (6 - 0)^2]
OB = √[6^2 + 6^2]
OB = √[36 + 36]
OB = √72
OB = 6√2
Now, let's calculate the ratio of the distance between the image of A and the origin to the distance between A and the origin. Since the dilation is centered at the origin, the image of A will be on the same line passing through the origin.
Image of A = (Scale factor * x-coordinate of A, Scale factor * y-coordinate of A)
Image of A = (Scale factor * 9, Scale factor * 9)
The distance between the origin and the image of A can be found using the distance formula:
Image of OA = √[(Scale factor * 9 - 0)^2 + (Scale factor * 9 - 0)^2]
Image of OA = √[(9Scale factor)^2 + (9Scale factor)^2]
Image of OA = √[81Scale factor^2 + 81Scale factor^2]
Image of OA = √[(81 + 81)Scale factor^2]
Image of OA = √[162Scale factor^2]
Image of OA = √162Scale factor
Now, let's set up the ratio:
Scale factor = (Image of OA) / OA
Scale factor = (√162Scale factor) / (9√2)
Scale factor = Scale factor / (9√2 / √162)
Scale factor = Scale factor / (9 / √2 * √162)
Scale factor = Scale factor / (9 / √(2 * 162))
Scale factor = Scale factor / (9 / √(324))
Scale factor = Scale factor / (9 / 18)
Scale factor = Scale factor * 18 / 9
Scale factor = Scale factor * 2
Thus, the scale factor is equal to 2.
Therefore, the correct response is 2.