Given the vertices of ∆ABC are A (2,-5), B (-4,6) and C (3,1), find the vertices following each of the transformations FROM THE ORIGINAL vertices: a. Rx = 3 b. T<3,-6> c. r(90◦, o)
To find the vertices after each transformation, we need to apply the transformation to each of the original coordinates of the vertices.
a. Reflection Rx = 3:
This reflection is done over the x-axis.
To reflect a point over the x-axis, we keep the x-coordinate the same, but change the sign of the y-coordinate.
For vertex A(2, -5):
After reflecting over the x-axis, the y-coordinate becomes -(-5) = 5.
So the new coordinates of vertex A are (2, 5).
Similarly, for vertex B (-4, 6):
After reflecting over the x-axis, the y-coordinate becomes -6.
So the new coordinates of vertex B are (-4, -6).
And for vertex C (3, 1):
After reflecting over the x-axis, the y-coordinate becomes -1.
So the new coordinates of vertex C are (3, -1).
Therefore, the new vertices after the reflection Rx = 3 are A' (2, 5), B' (-4, -6), and C' (3, -1).
b. Translation T<3, -6>:
This translation moves each point of the triangle by adding the given x and y values.
For vertex A (2, -5):
Adding the x- and y-values of the translation, the new coordinates of vertex A are (2 + 3, -5 - 6) = (5, -11).
For vertex B (-4, 6):
The new coordinates of vertex B are (-4 + 3, 6 - 6) = (-1, 0).
For vertex C (3, 1):
The new coordinates of vertex C are (3 + 3, 1 - 6) = (6, -5).
Therefore, the new vertices after the translation T<3, -6> are A' (5, -11), B' (-1, 0), and C' (6, -5).
c. Rotation r(90°, o):
This rotation is done counterclockwise by the given angle, 90°, with respect to the origin (0,0).
To rotate a point counterclockwise by 90°, we swap the x and y coordinates and negate the new x-coordinate.
For vertex A (2, -5):
Swapping and negating the new coordinates, the new coordinates of vertex A are (-(-5), 2) = (5, 2).
For vertex B (-4, 6):
The new coordinates of vertex B are (-6, -4).
For vertex C (3, 1):
The new coordinates of vertex C are (1, -3).
Therefore, the new vertices after the rotation r(90°, o) are A' (5, 2), B' (-6, -4), and C' (1, -3).