Question

Triangle XYZ has vertices at X(−5, −5), Y(2, 2), Z(2, −4).

Part A: If triangle XYZ is rotated 90° counterclockwise about the origin, determine the coordinates of the vertices for the image, triangle X′Y′Z′. Show each step of your work. (4 points)

Part B: If triangle XYZ is rotated 270° clockwise about the origin, determine the coordinates of the vertices for the image, triangle X′Y′Z′. Show each step of your work. (4 points)

Part C: Describe the similarities and differences between the two rotations in parts A and B. (4 points)

Answer 1

To rotate the given triangle 90° counterclockwise and 270° clockwise about the origin, we need to switch the x and y-coordinates and negate either the x or y-coordinates for each vertex. The coordinates of the image triangle for the counterclockwise rotation are (5,5), (2,2), and (-2,-4). The coordinates of the image triangle for the clockwise rotation are (-2,-2), (-2,2), and (2,4). The rotations are similar in preserving shape and size, but differ in which coordinates are negated and the direction of rotation.

Part A:

To rotate a point 90° counterclockwise about the origin on the coordinate plane, we need to switch the x and y-coordinates and negate the new x-coordinates.

Applying this to the given vertices:

X' = -(-5) = 5, Y' = -(-5) = 5

Z' = -(2) = -2

Therefore, the coordinates of the image triangle X'Y'Z' are X'(5, 5), Y'(2, 2), and Z'(-2, -4).

Part B:

To rotate a point 270° clockwise about the origin on the coordinate plane, we also need to switch the x and y-coordinates and negate the new y-coordinates.

Applying this to the given vertices:

X' = -(2) = -2, Y' = -(2) = -2

Z' = -(-4) = 4

Therefore, the coordinates of the image triangle X'Y'Z' are X'(-2, -2), Y'(-2, 2), and Z'(2, 4).

Part C:

The similarities between the two rotations are:

Both rotations preserve the shape and size of the triangle.

Both rotations involve switching the x and y-coordinates.

The differences between the two rotations are:

In Part A, the x-coordinates were negated, while in Part B, the y-coordinates were negated.

Part A rotates the triangle counterclockwise, while Part B rotates the triangle clockwise.