Triangle ABC has a vertices point of A(2,1), point B (4,3), and a 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' after a dilation with a scale factor of 7 and a center of dilation at the origin (0, 0), we multiply the x and y coordinates of A by the scale factor.
The x-coordinate of A' can be found by multiplying the x-coordinate of A by 7: x-coordinate of A' = 7 * 2 = 14.
The y-coordinate of A' can be found by multiplying the y-coordinate of A by 7: y-coordinate of A' = 7 * 1 = 7.
Therefore, the coordinates of A' are (14, 7).
We can follow the same process to find the coordinates of B' and C'.
For B:
The x-coordinate of B' = 7 * 4 = 28.
The y-coordinate of B' = 7 * 3 = 21.
Therefore, the coordinates of B' are (28, 21).
For C:
The x-coordinate of C' = 7 * -2 = -14.
The y-coordinate of C' = 7 * 5 = 35.
Therefore, the coordinates of C' are (-14, 35).
To summarize:
A' = (14, 7)
B' = (28, 21)
C' = (-14, 35)
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), you can follow these steps:
1. Calculate the distance between each vertex and the origin using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For point A(2,1):
Distance_A = sqrt((2 - 0)^2 + (1 - 0)^2) = sqrt(2^2 + 1^2) = sqrt(4 + 1) = sqrt(5)
For point B(4,3):
Distance_B = sqrt((4 - 0)^2 + (3 - 0)^2) = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5
For point C(-2,5):
Distance_C = sqrt((-2 - 0)^2 + (5 - 0)^2) = sqrt((-2)^2 + 5^2) = sqrt(4 + 25) = sqrt(29)
2. Multiply the x-coordinate and y-coordinate of each vertex by the scale factor (7) individually:
For point A(2,1):
A' = (7 * 2, 7 * 1) = (14, 7)
For point B(4,3):
B' = (7 * 4, 7 * 3) = (28, 21)
For point C(-2,5):
C' = (7 * -2, 7 * 5) = (-14, 35)
Therefore, the coordinates of A' are (14, 7), the coordinates of B' are (28, 21), and the coordinates of C' are (-14, 35).