Question

part c
a rotation 90° about the origin, then a reflection across x = 0.
t: ( )

Explanation:

To solve the problem, we need to know the original coordinates of point \( T \). Since the original coordinates are not provided in the image, we'll assume a general point \( T(x, y) \) and apply the transformations step by step.

Step 1: Rotation \( 90^\circ \) about the origin

The rule for a \( 90^\circ \) counterclockwise rotation about the origin is:
\( (x, y)
ightarrow (-y, x) \).

If we assume a clockwise rotation (common in some contexts), the rule is:
\( (x, y)
ightarrow (y, -x) \).

For clarity, we’ll use the counterclockwise rotation rule (standard in many curricula) unless specified otherwise.

Step 2: Reflection across \( x = 0 \) (the \( y \)-axis)

The rule for reflection across the \( y \)-axis (\( x = 0 \)) is:
\( (a, b)
ightarrow (-a, b) \).

Example: Let’s assume \( T \) has coordinates \( (x, y) \)

1. After \( 90^\circ \) counterclockwise rotation:

\( T \) becomes \( (-y, x) \).

1. After reflection across \( x = 0 \):

Apply the reflection rule to \( (-y, x) \):
\( (-(-y), x) = (y, x) \).

If \( T \) is a specific point (e.g., \( T(2, 1) \)):

1. Rotation \( 90^\circ \) counterclockwise: \( (2, 1)

ightarrow (-1, 2) \).

1. Reflection across \( x = 0 \): \( (-1, 2)

ightarrow (1, 2) \).

If \( T \) is \( (x, y) \), the final coordinates of \( T' \) (after both transformations) are \( \boldsymbol{(y, x)} \) (for counterclockwise rotation) or \( \boldsymbol{(-y, -x)} \) (for clockwise rotation).

Since the original coordinates of \( T \) are not given, we need that information to compute the exact coordinates of \( T' \). If you provide the original coordinates of \( T \), we can substitute and find \( T' \) directly.

Answer:

To solve the problem, we need to know the original coordinates of point \( T \). Since the original coordinates are not provided in the image, we'll assume a general point \( T(x, y) \) and apply the transformations step by step.

Step 1: Rotation \( 90^\circ \) about the origin

The rule for a \( 90^\circ \) counterclockwise rotation about the origin is:
\( (x, y)
ightarrow (-y, x) \).

If we assume a clockwise rotation (common in some contexts), the rule is:
\( (x, y)
ightarrow (y, -x) \).

For clarity, we’ll use the counterclockwise rotation rule (standard in many curricula) unless specified otherwise.

Step 2: Reflection across \( x = 0 \) (the \( y \)-axis)

The rule for reflection across the \( y \)-axis (\( x = 0 \)) is:
\( (a, b)
ightarrow (-a, b) \).

Example: Let’s assume \( T \) has coordinates \( (x, y) \)

1. After \( 90^\circ \) counterclockwise rotation:

\( T \) becomes \( (-y, x) \).

1. After reflection across \( x = 0 \):

Apply the reflection rule to \( (-y, x) \):
\( (-(-y), x) = (y, x) \).

If \( T \) is a specific point (e.g., \( T(2, 1) \)):

1. Rotation \( 90^\circ \) counterclockwise: \( (2, 1)

ightarrow (-1, 2) \).

1. Reflection across \( x = 0 \): \( (-1, 2)

ightarrow (1, 2) \).

If \( T \) is \( (x, y) \), the final coordinates of \( T' \) (after both transformations) are \( \boldsymbol{(y, x)} \) (for counterclockwise rotation) or \( \boldsymbol{(-y, -x)} \) (for clockwise rotation).

Since the original coordinates of \( T \) are not given, we need that information to compute the exact coordinates of \( T' \). If you provide the original coordinates of \( T \), we can substitute and find \( T' \) directly.