triangle ABC has vertices point A(4,3) and point C(-2,5) find the coordinates of A B C after the dilation with a scale factor of 7 an centerpoint of dilation at the origin 0,0
To find the coordinates of points A, B, and C after the dilation with a scale factor of 7 and a center point at the origin (0,0), you can use the following formula:
New_x = ScaleFactor * (Old_x - Center_x) + Center_x
New_y = ScaleFactor * (Old_y - Center_y) + Center_y
So, first, we find the coordinates of point B.
The midpoint of AC can be found using the formula:
Midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2 )
Midpoint = ( (4 + (-2)) / 2, (3 + 5) / 2 )
Midpoint = ( 2 / 2, 8 / 2 )
Midpoint = ( 1, 4 )
Now, we can find the coordinates of point B by reflecting point A across the midpoint.
New_x_B = ScaleFactor * (Old_x_A - Midpoint_x) + Midpoint_x
New_y_B = ScaleFactor * (Old_y_A - Midpoint_y) + Midpoint_y
New_x_B = 7 * (4 - 1) + 1
New_x_B = 7 * 3 + 1
New_x_B = 21 + 1
New_x_B = 22
New_y_B = 7 * (3 - 4) + 4
New_y_B = 7 * (-1) + 4
New_y_B = -7 + 4
New_y_B = -3
Therefore, the coordinates of point B after the dilation are (22, -3).
Now, we can find the coordinates of point C after the dilation using the same formula.
New_x_C = ScaleFactor * (Old_x_C - Center_x) + Center_x
New_y_C = ScaleFactor * (Old_y_C - Center_y) + Center_y
New_x_C = 7 * (-2 - 1) + 1
New_x_C = 7 * (-3) + 1
New_x_C = -21 + 1
New_x_C = -20
New_y_C = 7 * (5 - 4) + 4
New_y_C = 7 * 1 + 4
New_y_C = 7 + 4
New_y_C = 11
Therefore, the coordinates of point C after the dilation are (-20, 11).
Summarizing the results, the coordinates of points A, B, and C after the dilation with a scale factor of 7 and a center point at the origin (0,0) are:
A(4,3) -> A'(4,3) (no change)
B(22,-3) -> B'(22,-3)
C(-2,5) -> C'(-20,11)