Question

a line segment has endpoints at \\((3, 2)\\) and \\((2, -3)\\). which reflection will produce an image with endpoints at \\((3, -2)\\) and \\((2, 3)\\)?

a reflection of the line segment across the \\(x\\)-axis
a reflection of the line segment across the \\(y\\)-axis
a reflection of the line segment across the line \\(y = x\\)
a reflection of the line segment across the line \\(y = -x\\)

Explanation:

Identify the given endpoints

We are given a line segment with original endpoints:

$$ A(3, 2) \quad \text{and} \quad B(2, -3) $$

The reflected image has endpoints:

$$ A'(3, -2) \quad \text{and} \quad B'(2, 3) $$

Analyze coordinate changes

Let's compare the coordinates of each original point with its image:

• For the first endpoint:

$$ (3, 2) ightarrow (3, -2) $$

Here, the \(x\)-coordinate remains unchanged, while the \(y\)-coordinate is negated:

$$ (x, y) ightarrow (x, -y) $$

• For the second endpoint:

$$ (2, -3) ightarrow (2, 3) $$

Similarly, the \(x\)-coordinate remains unchanged, while the \(y\)-coordinate is negated:

$$ (x, y) ightarrow (x, -y) $$

Determine the transformation rule

The transformation mapping each point \((x, y)\) to \((x, -y)\) represents a reflection across the \(x\)-axis.
This is because reflecting a point across the horizontal \(x\)-axis keeps its horizontal distance from the \(y\)-axis the same but flips its vertical position.

Match with the options

Let's evaluate the given choices:

• Option 1: a reflection of the line segment across the \(x\)-axis. This matches our rule \((x, y)

ightarrow (x, -y)\).

• Option 2: a reflection of the line segment across the \(y\)-axis. This corresponds to \((x, y)

ightarrow (-x, y)\).

• Option 3: a reflection of the line segment across the line \(y = x\). This corresponds to \((x, y)

ightarrow (y, x)\).

• Option 4: a reflection of the line segment across the line \(y = -x\). This corresponds to \((x, y)

ightarrow (-y, -x)\).

Answer:

• a reflection of the line segment across the x-axis (Correct answer)
• a reflection of the line segment across the y-axis
• a reflection of the line segment across the line y = x
• a reflection of the line segment across the line y = -x