Determine the scale factor used in the dilation of line segment AB , centered at the origin.
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 and the dilated line segment.
Let's assume that AB is the original line segment and A'B' is the dilated line segment.
The scale factor, denoted as k, is the ratio of the length of the dilated line segment to the length of the original line segment. Mathematically, we can express this as:
k = Length of A'B' / Length of AB
Since the line segment is centered at the origin, we can use the distance formula to find the lengths of AB and A'B'. The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's say the coordinates of point A are (x1, y1) and the coordinates of point B are (x2, y2). Similarly, the coordinates of point A' are (x1', y1') and the coordinates of point B' are (x2', y2').
Using the distance formula, we can calculate the lengths of AB and A'B':
Length of AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Length of A'B' = sqrt((x2' - x1')^2 + (y2' - y1')^2)
Now, we can substitute the lengths into the scale factor formula:
k = Length of A'B' / Length of AB
k = sqrt((x2' - x1')^2 + (y2' - y1')^2) / sqrt((x2 - x1)^2 + (y2 - y1)^2)
The scale factor is the ratio of the lengths of the original line segment and the dilated line segment, and it depends on the specific coordinates of A, B, A', and B'. You would need to provide the coordinates of these points in order for us to calculate the scale factor specifically for your problem.
so what's the scale factor
I apologize for the confusion. Without the specific coordinates of the points A, B, A', and B', I cannot calculate the scale factor. Please provide the coordinates of these points, and I will be able to calculate the scale factor for you.
oh sorry there's (2, 3) (4, 1) (6, 9) (12, 3)
Thank you for providing the coordinates.
Using the distance formula, we can calculate the lengths of AB and A'B':
Length of AB = sqrt((4 - 2)^2 + (1 - 3)^2) = sqrt(4 + 4) = sqrt(8)
Length of A'B' = sqrt((12 - 6)^2 + (3 - 9)^2) = sqrt(36 + 36) = sqrt(72)
Now, we can substitute the lengths into the scale factor formula:
k = Length of A'B' / Length of AB
k = sqrt(72) / sqrt(8)
To simplify this, we can write it as:
k = sqrt(72 / 8)
k = sqrt(9)
k = 3
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 3.
Determine the scale factor used in the dilation of line AB , with a center point of dilation at the origin.
the coordinates are (4, -8) (2, -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 and the dilated line segment.
Let's assume that AB is the original line segment and A'B' is the dilated line segment.
The scale factor, denoted as k, is the ratio of the length of the dilated line segment to the length of the original line segment. Mathematically, we can express this as:
k = Length of A'B' / Length of AB
Since the line segment is centered at the origin, we can use the distance formula to find the lengths of AB and A'B'. The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's say the coordinates of point A are (x1, y1) and the coordinates of point B are (x2, y2). Similarly, the coordinates of point A' are (x1', y1') and the coordinates of point B' are (x2', y2').
Using the distance formula, we can calculate the lengths of AB and A'B':
Length of AB = sqrt((2 - 4)^2 + (-4 - (-8))^2)
= sqrt((-2)^2 + 4^2)
= sqrt(4 + 16)
= sqrt(20)
Length of A'B' = sqrt((2' - 4')^2 + (-4' - (-8'))^2)
= sqrt((-2')^2 + 4'^2)
= sqrt(4' + 16')
= sqrt(20')
Now, we can substitute the lengths into the scale factor formula:
k = Length of A'B' / Length of AB
k = sqrt(20') / sqrt(20)
As the point (4, -8) maps to the point (2, -4) under the dilation centered at the origin, the scale factor must be 2.
Triangle ABC has vertices point A(2,1) , point B(4,3) , and point C(−2,5) . Find the coordinates of A′ , B′ , and C′ after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0) .
To find the coordinates of A', B', and C' after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0), we can use the dilation formula:
A' = (7 * x-coordinate of A, 7 * y-coordinate of A)
B' = (7 * x-coordinate of B, 7 * y-coordinate of B)
C' = (7 * x-coordinate of C, 7 * y-coordinate of C)
Using the given coordinates, we have:
A' = (7 * 2, 7 * 1)
= (14, 7)
B' = (7 * 4, 7 * 3)
= (28, 21)
C' = (7 * -2, 7 * 5)
= (-14, 35)
Therefore, the coordinates of A', B', and C' after the dilation are A'(14, 7), B'(28, 21), and C'(-14, 35) respectively.