To solve the problem of rotating △ABC 90 degrees counterclockwise about the origin and then reflecting it across the y-axis, let’s find the coordinates step by step. I'll assume we have the original coordinates for triangle ABC as follows:
- A(x₁, y₁)
- B(x₂, y₂)
- C(x₃, y₃)
Question 1: Rotate △ABC 90 degrees counterclockwise
The formula for rotating a point (x, y) counterclockwise by 90 degrees about the origin is:
- (x, y) → (-y, x)
So we will apply this transformation to each vertex:
-
For point A (x₁, y₁): A' = (-y₁, x₁)
-
For point B (x₂, y₂): B' = (-y₂, x₂)
-
For point C (x₃, y₃): C' = (-y₃, x₃)
Let's say the coordinates of the original triangle are:
- A(1, 2)
- B(3, 4)
- C(5, 6)
Then the new coordinates after rotation will be:
- A' = (-2, 1)
- B' = (-4, 3)
- C' = (-6, 5)
Question 2: Reflect △A′B′C′ across the y-axis
The formula for reflecting a point (x, y) across the y-axis is:
- (x, y) → (-x, y)
Applying this reflection to A', B', and C':
- A'' = (2, 1) (A' = (-2, 1))
- B'' = (4, 3) (B' = (-4, 3))
- C'' = (6, 5) (C' = (-6, 5))
Coordinates after reflection across the y-axis
- A''(2, 1)
- B''(4, 3)
- C''(6, 5)
Question 3: True or False? We need to determine whether △ABC is congruent to △A"B"C".
Since the first transformation was a rotation and the second transformation was a reflection, both of which preserve the shape and size of the triangle, we can conclude that:
- True: Since the triangles are congruent after the transformations.
So the answers would be:
Question 1:
A′ ( -2, 1)
B′ ( -4, 3)
C′ ( -6, 5)
Question 2:
A'' ( 2, 1)
B'' ( 4, 3)
C'' ( 6, 5)
Question 3: True
1 answer