Each dot’s mirror twin sits the same distance beyond the line
Every corner has a twin
Half of a figure is drawn on a dot grid, beside a dashed mirror line. The task is to complete the figure so that the mirror line is a line of symmetry: folded along the line, the new half must land exactly on the half that is drawn.
For that, every corner of the drawn half needs a twin on the other side of the line. The twin is on the same row as the corner and the same distance from the line, but on the opposite side.
The left half is drawn, and it meets the dashed mirror line at two points.
Count, cross, count
Take one corner at a time. Count the gaps between dots from the corner to the mirror line, along its row. The top-left corner is 3 gaps to the left of the line. Cross the line and count the same number of gaps on the other side: its twin is 3 gaps to the right of the line, on the same row.
Do the same for every corner. A corner 1 gap from the line has its twin 1 gap beyond it, and a point that sits on the mirror line is its own twin. Then join the twins in the same order as the corners they copy.
The corner 3 gaps to the left of the line has its twin 3 gaps to the right, on the same row.
The completed figure: each corner on the left has a twin on the right, the same number of gaps from the line.
A mirror line across the grid
The mirror line can also run across the grid. Then count the gaps up and down instead. A corner 2 gaps above the line has its twin 2 gaps below it, in the same column.
The two top corners are 2 gaps above the line, and their twins are 2 gaps below it. The two corners on the line are their own twins.
Flip it, do not slide it
Two slips are common. Counting from the edge of the grid, or from the middle of the drawn half, instead of from the mirror line, puts the twin in the wrong place. And a mirror moves a corner only across the line, never along it: the twin of a corner is always on the same row when the line runs up and down.
The new half faces the other way from the drawn half. A copy slid across the line would face the same way, and it would not fold onto the drawn half.
Symmetry and grids · A Square Sheet Folded Once, with One Hole Punched
QuestionA square sheet of paper is folded in half, so that its left edge lands on its right edge. A hole is punched through both layers, 2 cm from the fold and 3 cm from the top edge. (a) When the sheet is opened out, how many holes are there? (b) How far apart are the holes?
1.The fold lays the left half of the sheet exactly on the right half, so the folded paper is 2 layers thick.
Folded left onto right, the sheet is 2 layers thick. 2.(a) One punch through 2 layers makes 2 holes.
(a) One punch through 2 layers makes 2 holes. 3.Open the sheet. The fold line is a line of symmetry, so the hole in the right half has a twin in the left half. The twin is also 2 cm from the fold, at the same height, 3 cm from the top.
Opened, the fold is a line of symmetry: each hole is 2 cm from it, 3 cm from the top. 4.(b) The holes are 2 + 2 = 4 cm apart. Check: each hole is 2 cm from the fold, one on each side of it, so the fold is exactly halfway between them.
(b) The holes are 2 + 2 = 4 cm apart.
Answer: (a) 2 holes; (b) 4 cm apart
Watch out for
- Answering 2 cm apart. That is the distance from one hole to the fold, which is only half of the gap. The other hole is 2 cm from the fold on the other side, so the holes are 2 + 2 = 4 cm apart.
- Answering 1 hole. The folded paper is 2 layers thick and the punch goes through both, so when the sheet is opened there is a hole in each half.
Symmetry and grids · A Square Folded Along a Diagonal, with a Corner Cut Off
QuestionA square sheet of paper ABCD is folded along the diagonal AC, so that corner B lands on corner D. A small triangle is cut off the folded corner, where B and D lie together. (a) When the sheet is opened out, how many corners of the square have been cut off? (b) How many sides does the shape have now?
1.Folding along the diagonal AC lays triangle ABC exactly on triangle ADC, so corner B lands on corner D.
Folded along AC, corner B lies exactly on corner D. 2.The cut goes through both layers at the corner where B and D lie together, so it cuts off both corners at once.
One cut through both layers removes both corners at once. 3.(a) When the sheet is opened, corners B and D have been cut off. Corners A and C are at the two ends of the fold line, away from the cut, so they are still there. That is 2 corners cut off.
(a) Opened, B and D are cut off. A and C, at the ends of the fold, are still there. 4.Cutting off a corner leaves the two sides at that corner in place, only shorter, and makes one new short side between them. So each cut adds 1 side.
Each cut leaves the two sides at the corner, only shorter, and adds one short new side. 5.(b) The square had 4 sides, and the two cuts add 2 more, so the shape has 4 + 2 = 6 sides.
(b) The shape has 4 + 2 = 6 sides.
Answer: (a) 2 corners; (b) 6 sides
Watch out for
- Answering 1 corner. At the folded corner the paper is 2 layers thick, with B on top of D, so the one cut goes through both layers and removes both corners.
- Answering 4 sides, as if cutting off a corner took a side away. The two sides at the corner only get shorter, and the cut makes one new side between them, so each cut corner adds a side: 4 + 2 = 6.
Your turn
1.The figure’s open corner reflects to which dot?