Depth · one concept from first principles · step 2 of 4

Fold or half-turn overlap reveals symmetry

You want to use $4$ right-triangle tiles, rotating and joining them, to make a line-symmetric figure. Draw the remaining $2$ tiles in the empty cells to complete a line-symmetric figure.

Each tile is a right triangle made by cutting a unit square along one diagonal (half shaded, half white). The figure is a $2 \times 2$ grid, and the two bottom cells already have tiles in them. In both the bottom-left and bottom-right cells, the shaded right triangle fills the lower-left half of the cell (the diagonal runs from the top-left corner to the bottom-right corner). The two top cells are empty.

Geometry

Your answer

How to solve

Strategy Draw a Diagram — Symmetry is something you see, so I draw the grid and test fold lines. Imagining (or actually) folding the paper along a candidate line and checking whether the shaded halves land on top of each other is exactly the 'half-turn / fold overlap' idea of this lesson. I sketch the two bottom tiles, pick a fold line, and figure out what the top tiles must be so the fold matches.

1STEP 1

Describe each filled bottom tile

Both bottom cells are shaded in the lower-left triangle, the slant running top-left to bottom-right.

2STEP 2

Try the vertical fold line and reject it

A vertical fold turns a lower-left triangle into a lower-right one, so the bottom row cannot match.

3STEP 3

Use the horizontal fold line instead

Fold along the horizontal middle instead, sending each bottom tile straight up into an empty cell.

4STEP 4

Mirror each bottom tile upward to find the top tiles

Flipping a lower-left triangle over a horizontal crease gives an upper-left triangle.

5STEP 5

Check the finished figure folds onto itself

Folded along that middle line, the top shading lands exactly on the bottom shading.

Answer

Fill BOTH top cells with a shaded upper-left triangle (slant from the bottom-left corner to the top-right corner). The figure is then line-symmetric about the horizontal middle line.

Two tiles are used, one per empty cell, as required. The completed shaded shape (top row pointing up, bottom row pointing down) is a mirror pair across the horizontal center line, so it genuinely has a line of symmetry. The total shaded area is 4 half-squares = 2 squares, balanced top and bottom.

Takeaway

Fold the paper in half: whatever you draw on top should land right on the matching shape underneath.

Describe each filled bottom tile
Try the vertical fold line and reject it
Use the horizontal fold line instead
Mirror each bottom tile upward to find the top tiles
Check the finished figure folds onto itself

Where next?

Level upsuggested

Overlapping congruent shapes leave congruent leftovers Symmetry axis or center bisects corresponding-point segment

Step back and solidifysuggested

Even number of flips returns original Flip across a line creates symmetry

3 reasoning problems grow out of this type — the bridge toward competition →

A parent dashboard for the family lives at sensimlab.com.

The same shape shows up later on the AMC, as Paper Fold-and-Cut Symmetry. Why the shapes repeat →