Reflect triangle ABC over the y-axis and then rotate triangle 90° clockwise around the center. Plot triangle DEF.
A(2,3)
B(4,3)
C(2,6)
Xmin:
-10
Xmax:
10
Ymin:
-10
Ymax:
10
To reflect triangle ABC over the y-axis, we simply change the sign of the x-coordinates.
New coordinates:
A'(-2, 3)
B'(-4, 3)
C'(-2, 6)
To rotate triangle ABC 90° clockwise around the center (which is the midpoint of the line segment AB), we first need to find the coordinates of the midpoint.
Midpoint of AB:
[(x1 + x2) / 2, (y1 + y2) / 2]
[(2 + 4) / 2, (3 + 3) / 2]
[6 / 2, 6 / 2]
[3, 3]
Now we can rotate each point by 90° clockwise around the center.
Rotation formula:
x' = x_center + (x - x_center) * cos(theta) - (y - y_center) * sin(theta)
y' = y_center + (x - x_center) * sin(theta) + (y - y_center) * cos(theta)
Substituting the values into the formula:
For point A':
x_center = 3, y_center = 3, x = -2, y = 3, theta = -90°
x' = 3 + (-2 - 3) * cos(-90°) - (3 - 3) * sin(-90°)
x' = 3 + (-5) * 0 - 0
x' = 3
y' = 3 + (-2 - 3) * sin(-90°) + (3 - 3) * cos(-90°)
y' = 3 + (-5) * 1 + 0
y' = 3 - 5
y' = -2
So, point A' = (3, -2)
Now, repeating the same process for points B and C:
For point B':
x_center = 3, y_center = 3, x = -4, y = 3, theta = -90°
x' = 3 + (-4 - 3) * cos(-90°) - (3 - 3) * sin(-90°)
x' = 3 + (-7) * 0 - 0
x' = 3
y' = 3 + (-4 - 3) * sin(-90°) + (3 - 3) * cos(-90°)
y' = 3 + (-7) * 1 + 0
y' = 3 - 7
y' = -4
So, point B' = (3, -4)
For point C':
x_center = 3, y_center = 3, x = -2, y = 6, theta = -90°
x' = 3 + (-2 - 3) * cos(-90°) - (6 - 3) * sin(-90°)
x' = 3 + (-5) * 0 - 3
x' = 3 - 3
x' = 0
y' = 3 + (-2 - 3) * sin(-90°) + (6 - 3) * cos(-90°)
y' = 3 + (-5) * 1 + 3 * 0
y' = 3 - 5
y' = -2
So, point C' = (0, -2)
The corresponding plot for triangle DEF is:
D(3, -2)
E(3, -4)
F(0, -2)
Here are the plot coordinates:
Triangle ABC: (2,3), (4,3), (2,6)
Triangle DEF: (3,-2), (3,-4), (0,-2)
Note: The given x and y limits do not affect the reflection and rotation of the triangle, but they provide a range for the plot coordinates.