Perpendicular and parallel lines on squared paper
On squared paper, draw AB going 6 squares right. Then draw XY ⊥ AB and CD ∥ AB.
On squared paper, draw AB going 5 squares up. Then draw XY ⊥ AB and CD ∥ AB.
On squared paper, draw AB going 6 squares right and 2 squares up. Then draw XY ⊥ AB and CD ∥ AB.
On squared paper, draw AB going 4 squares right and 4 squares up. Then draw XY ⊥ AB and CD ∥ AB.
On squared paper, draw AB going 6 squares right and 3 squares down. Then draw XY ⊥ AB and CD ∥ AB.
1. Start at a corner of the squares, A, and go 6 squares right to B. Join AB with a ruler.
2. AB runs along a line of the squares, so the line of squares across it is perpendicular to it. From O on AB, go 2 squares up to X, and the opposite way to Y.
3. Join XY: it crosses AB at a right angle, XY ⊥ AB.
4. Mark C, 3 squares down from A, and from C go the same way as AB, 6 squares right, to D. Join CD: CD ∥ AB.
1. Start at a corner of the squares, A, and go 5 squares up to B. Join AB with a ruler.
2. AB runs along a line of the squares, so the line of squares across it is perpendicular to it. From O on AB, go 2 squares left to X, and the opposite way to Y.
3. Join XY: it crosses AB at a right angle, XY ⊥ AB.
4. Mark C, 3 squares right from A, and from C go the same way as AB, 5 squares up, to D. Join CD: CD ∥ AB.
1. Start at a corner of the squares, A, and go 6 squares right and 2 squares up (2 times 3 squares right and 1 square up) to B. Join AB with a ruler.
2. Each step of AB goes 3 squares right and 1 square up; a perpendicular turns it round and goes 1 square left and 3 squares up. From O on AB, go 1 square left and 3 squares up to X, and the opposite way to Y.
3. Join XY: it crosses AB at a right angle, XY ⊥ AB.
4. Mark C, 3 squares down from A, and from C go the same way as AB, 6 squares right and 2 squares up, to D. Join CD: CD ∥ AB.
1. Start at a corner of the squares, A, and go 4 squares right and 4 squares up (4 times 1 square right and 1 square up) to B. Join AB with a ruler.
2. Each step of AB goes 1 square right and 1 square up; a perpendicular turns it round and goes 1 square left and 1 square up. From O on AB, go 2 squares left and 2 squares up to X, and the opposite way to Y.
3. Join XY: it crosses AB at a right angle, XY ⊥ AB.
4. Mark C, 2 squares right and 2 squares down from A, and from C go the same way as AB, 4 squares right and 4 squares up, to D. Join CD: CD ∥ AB.
1. Start at a corner of the squares, A, and go 6 squares right and 3 squares down (3 times 2 squares right and 1 square down) to B. Join AB with a ruler.
2. Each step of AB goes 2 squares right and 1 square down; a perpendicular turns it round and goes 1 square right and 2 squares up. From O on AB, go 1 square right and 2 squares up to X, and the opposite way to Y.
3. Join XY: it crosses AB at a right angle, XY ⊥ AB.
4. Mark C, 3 squares down from A, and from C go the same way as AB, 6 squares right and 3 squares down, to D. Join CD: CD ∥ AB.
Why it works
On squared paper every line of the squares crosses the others at 90°. Going a squares across and b up, then turning the picture a quarter turn, gives b squares back and a up: the same line turned through 90°, so it is perpendicular. A line that takes the same steps from another point keeps the same distance from the first one all along: it is parallel.