Reflecting Letter C Across X- and Y-Axes

Problem

Reflect a block letter $C$ on the Cartesian plane. Use the original points of the letter $C$, then find its reflection across the $x$-axis and across the $y$-axis.

Step 1: Choose Points for the Original Figure

To begin the reflection task, the letter $C$ needs fixed points on the Cartesian plane before any flipping can happen.

A neat block letter $C$ can use the following points:

$$A(-4,4),\ B(2,4),\ C(2,2),\ D(-2,2),\ E(-2,-2),\ F(2,-2),\ G(2,-4),\ H(-4,-4)$$

Step 2: Connect the Points into Letter C

With the original coordinates chosen, the points are connected in order to form a block letter $C$ instead of a loose set of dots.

Connect the points as follows:

$$A \to B \to C \to D \to E \to F \to G \to H$$

This creates the original block letter $C$.

Step 3: Use the $x$-Axis Reflection Rule

Reflecting across the $x$-axis keeps each $x$-coordinate the same and changes each $y$-coordinate to its opposite.

The rule is:

$$(x,y) \to (x,-y)$$

Step 4: Reflect the Letter Across the $x$-Axis

Using the $x$-axis reflection rule, each point keeps its horizontal position but flips to the opposite vertical position.

The reflected points are:

$$A'(-4,-4),\ B'(2,-4),\ C'(2,-2),\ D'(-2,-2)$$ $$E'(-2,2),\ F'(2,2),\ G'(2,4),\ H'(-4,4)$$

Step 5: Use the $y$-Axis Reflection Rule

Reflecting across the $y$-axis changes each $x$-coordinate to its opposite and keeps each $y$-coordinate the same.

The rule is:

$$(x,y) \to (-x,y)$$

Step 6: Reflect the Letter Across the $y$-Axis

Using the $y$-axis reflection rule, each point keeps its vertical position but moves to the opposite horizontal position.

The reflected points are:

$$A''(4,4),\ B''(-2,4),\ C''(-2,2),\ D''(2,2)$$ $$E''(2,-2),\ F''(-2,-2),\ G''(-2,-4),\ H''(4,-4)$$

Step 7: Final Answer

The original letter $C$ has coordinates:

$$A(-4,4),\ B(2,4),\ C(2,2),\ D(-2,2),\ E(-2,-2),\ F(2,-2),\ G(2,-4),\ H(-4,-4)$$

The reflection across the $x$-axis is:

$$A'(-4,-4),\ B'(2,-4),\ C'(2,-2),\ D'(-2,-2),\ E'(-2,2),\ F'(2,2),\ G'(2,4),\ H'(-4,4)$$

The reflection across the $y$-axis is:

$$A''(4,4),\ B''(-2,4),\ C''(-2,2),\ D''(2,2),\ E''(2,-2),\ F''(-2,-2),\ G''(-2,-4),\ H''(4,-4)$$