Triangles DEF and D′E′F′ are shown on the coordinate plane below:

Triangle DEF and triangle D prime E prime F prime with ordered pairs at D negative 1, 6, at E 1, 3, at F 6, 3, at D prime negative 6, negative 1, at E prime negative 3, 1, at F prime negative 3, 6.

What rotation was applied to triangle DEF to create triangle D′E′F′?
(x, y) → (−y, x)
(x, y) → (y, −x)
(x, y) → (x, −y)
(x, y) → (−x, −y)

1 answer

To determine the rotation that maps triangle DEF to triangle D′E′F′, we will analyze the coordinates of both triangles:

Coordinates of Triangle DEF:
- D(-1, 6)
- E(1, 3)
- F(6, 3)
Coordinates of Triangle D′E′F′:
- D′(-6, -1)
- E′(-3, 1)
- F′(-3, 6)

We will apply the given transformations and see which one maps triangle DEF to triangle D′E′F′.

Transformation: (x, y) → (−y, x) (90-degree rotation counter-clockwise)
- D(-1, 6) → (−6, -1) (matches D′)
- E(1, 3) → (−3, 1) (matches E′)
- F(6, 3) → (−3, 6) (matches F′)

This transformation matches all three points D, E, and F to D′, E′, and F′ respectively.

Transformation: (x, y) → (y, −x) (90-degree rotation clockwise)
- D(-1, 6) → (6, 1) (does not match)

This transformation does not work.

Transformation: (x, y) → (x, −y) (Reflection across x-axis)
- D(-1, 6) → (-1, -6) (does not match)

This transformation does not work.

Transformation: (x, y) → (−x, −y) (Rotation by 180 degrees)
- D(-1, 6) → (1, -6) (does not match)

This transformation does not work.

Therefore, the rotation that maps triangle DEF to triangle D′E′F′ is:

(x, y) → (−y, x) (90-degree rotation counter-clockwise).