2001 AMC 10 Problem 5

A shape has a line of reflectional symmetry if you can fold it along some line and the two halves land exactly on top of each other. The line can be across, up and down, or slanted at $45^\circ$.

Go through the twelve shapes one at a time. The usual letter names are given to keep track.

Shapes that have a fold line:

1. The straight bar of $5$ squares (I): fold it down the middle the long way, or across the middle square.
2. The T shape: fold it down the middle of its stem.
3. The U shape: fold it down the middle, between the two raised ends.
4. The corner with two arms of $3$ squares each (V): fold along the slanted line through the corner square.
5. The staircase (W): fold along the slanted line through its middle square.
6. The plus sign (X): it has four fold lines.

Shapes that do not:

1. The F, the L (a bar of $4$ with one square on the end), the N (a row of $3$ and a row of $2$ shifted over), the P (a $2 \times 2$ block with one extra square), the Y (a bar of $4$ with one square sticking out from the second square), and the Z.
2. The Z is the tricky one. It looks balanced, but that is because it matches itself when you turn it halfway around. Flipping it over gives an S, which is a different shape, so it has no fold line.

Counting the first list gives $6$ shapes.

The answer is $\boxed{\textbf{(D)}\ 6}$.

With only twelve shapes, a careful check of each one is the fastest method. Two things to watch for: a shape can match itself after a turn without having any fold line (like Z or N), and some fold lines are slanted, not level or upright (like in V and W). Test all four directions: across, up and down, and both diagonals.

Check each shape for a mirror line, allowing horizontal, vertical, and diagonal axes.

Shapes with a line of symmetry:
- the straight bar (I): mirror along its length and across its middle;
- the T shape: mirror down its stem;
- the U shape: mirror between its two prongs;
- the L-shaped corner with equal arms of length $3$ (V): mirror along the diagonal through the corner;
- the staircase (W): mirror along the diagonal through the middle square;
- the plus sign (X): four mirror lines.

That is $6$ shapes.

Shapes without one: the F, the L (bar of four with one cell on the end), the N (offset $3$ and $2$), the P (a $2 \times 2$ block with a tail), the Y (bar of four with one cell off the second square), and the Z. The Z looks symmetric but only returns to itself under a half-turn, not a reflection.

The answer is $\boxed{\textbf{(D)}\ 6}$.