Grade 10 Math Puzzle 4: Choose a perpendicular direction using a half-plane

A free grade 10 math puzzle that teaches Choose a perpendicular direction using a half-plane. Try it first — the hint and the full solution open when you are ready.

Problem 4 · Choose a perpendicular direction using a half-plane · 4 points

Made by AI and not reviewed by a real person.

A square has one side from the drawn line’s y-axis crossing to its grid point with x = 1. Its interior is on the same side of the line as the origin. Which grid point is diagonally opposite the y-axis crossing?

a numbered coordinate grid (x from -3 to 3, y from -5 to 7, square cells) with the line y = 2x + 1 drawn and labelled: it crosses the y-axis at 1 and rises 2 for every 1 to the right; no other line

(3, 2)
(2, 0)
(3, 4)
(−2, 2)
(−1, 4)

Show a hint

That side could belong to a square on either side of the line. How can the origin help you choose between them?
The side from (0, 1) to (1, 3) moves right 1 and up 2. For the next side, swap the step sizes and reverse one direction; then decide which turn points toward the origin’s side.

Show the answer and solution

Answer: A — (3, 2)

From (0, 1) to (1, 3), the side moves (1, 2). An equal-length perpendicular move toward the origin’s side is (2, −1), which has slope −1/2. Adding it to (1, 3) gives the diagonally opposite corner (3, 2). The tempting (−1, 4) uses the opposite turn, placing the square on the wrong side of the line.

Grade 10 puzzle 4: Choose a perpendicular direction using a half-plane

Problem 4 · Choose a perpendicular direction using a half-plane · 4 points