Question

Rotate △ABC (A = 5, -1) (B = 5,-9) (C = 2,-7)
90 degrees (counterclockwise) about the origin to create △A′B′C′
. What are the coordinates of the following points?(3 points)
A′
(
,
)

B′
(
,
)

C′
(
,
)

Answer 1

Now reflect △A′B′C′
across the y-axis to create △A"B"C"
. What are the new coordinates?(3 points)
A"
(
,
)

B"
(
,
)

C"
(
,
)

Answer 2

No, use this coordinates.
(A = 5, -1) (B = 5,-9) (C = 2,-7)

Answer 3

True or False?

△ABC ≅ △A"B"C"
(1 point)
Responses

True
True

False

Answer 4

To rotate points \( A \), \( B \), and \( C \) 90 degrees counterclockwise about the origin, we can use the following transformation:

If a point \( (x, y) \) is rotated 90 degrees counterclockwise, its new coordinates \( (x', y') \) will be given by:

\[
(x', y') = (-y, x)
\]

Now, let's apply this transformation to each point:

1. **For point \( A(5, -1) \)**:
\[
A' = (-(-1), 5) = (1, 5)
\]

2. **For point \( B(5, -9) \)**:
\[
B' = (-(-9), 5) = (9, 5)
\]

3. **For point \( C(2, -7) \)**:
\[
C' = (-(-7), 2) = (7, 2)
\]

Thus, the new coordinates after rotation are:

- \( A' (1, 5) \)
- \( B' (9, 5) \)
- \( C' (7, 2) \)

In summary:
- A′ (1, 5)
- B′ (9, 5)
- C′ (7, 2)

Answer 5

To reflect points across the y-axis, we transform the coordinates \( (x, y) \) to \( (-x, y) \).

Let's apply this transformation to the points \( A' \), \( B' \), and \( C' \):

1. **For point \( A'(1, 5) \)**:
\[
A'' = (-1, 5)
\]

2. **For point \( B'(9, 5) \)**:
\[
B'' = (-9, 5)
\]

3. **For point \( C'(7, 2) \)**:
\[
C'' = (-7, 2)
\]

Thus, the new coordinates after the reflection are:

- \( A'' (-1, 5) \)
- \( B'' (-9, 5) \)
- \( C'' (-7, 2) \)

In summary:
- A" (-1, 5)
- B" (-9, 5)
- C" (-7, 2)

Answer 6

Apologies for the confusion! Let's reflect the original triangle points \( A(5, -1) \), \( B(5, -9) \), and \( C(2, -7) \) across the y-axis.

To reflect a point \( (x, y) \) across the y-axis, the new coordinates will be \( (-x, y) \).

1. **For point \( A(5, -1) \)**:
\[
A'' = (-5, -1)
\]

2. **For point \( B(5, -9) \)**:
\[
B'' = (-5, -9)
\]

3. **For point \( C(2, -7) \)**:
\[
C'' = (-2, -7)
\]

Thus, the new coordinates after the reflection across the y-axis are:

- \( A'' (-5, -1) \)
- \( B'' (-5, -9) \)
- \( C'' (-2, -7) \)

In summary:
- A" (-5, -1)
- B" (-5, -9)
- C" (-2, -7)

Answer 7

To determine whether \( \triangle ABC \) is congruent to \( \triangle A"B"C" \), we can use the properties of transformations.

1. **Original Triangle \( \triangle ABC \)**:
- Points:
- \( A(5, -1) \)
- \( B(5, -9) \)
- \( C(2, -7) \)

2. **Transformed Triangle \( \triangle A"B"C" \)**:
- Points:
- \( A''(-5, -1) \)
- \( B''(-5, -9) \)
- \( C''(-2, -7) \)

### Analysis
- The transformation that we applied (rotating \( 90^\circ \) counterclockwise around the origin and then reflecting across the y-axis) preserves distance and angles, which is the basis for congruence of triangles.
- Therefore, the transformations keep the sizes and shapes of the triangles the same.

### Conclusion
Since transformations (rotation and reflection) preserve congruence, \( \triangle ABC \) is indeed congruent to \( \triangle A"B"C" \).

**Answer:** **True**