Question 9 of 50advanced🔧 ApplyShort Answer3 marks

You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?

Correct Answer

There are infinitely many squares that can be drawn on the plane with points A and B on the boundary. Just as multiple circles can pass through two points by varying the radius and center along the perpendicular bisector, a square can also be resized and repositioned in numerous ways while keeping A and B on its boundary lines.

There are exactly 3 squares that can be drawn with A and B as the corners of the square. There are two cases to consider:

  1. A and B are adjacent corners: The other two corners are obtained by rotating segment AB by +90° and −90° about A and B respectively. This yields 2 squares (one on each side of line AB).

  2. A and B are diagonal (opposite) corners: The other two corners must be equidistant from A and B, so they lie on the perpendicular bisector of segment AB at distance |AB|/2 from the midpoint. These two points (one on each side of AB) together form exactly 1 square.

Total = 2 + 1 = 3 squares.

Exercise: Think and Reflect | Q: 5 | (Page 4)
For More Understanding

Explanation

The textbook context establishes that points equidistant from A and B lie on the perpendicular bisector of AB, and explores how many circles pass through two points. Applying this logic, if A and B are merely on the boundary, infinite squares are possible by varying size and position. If they are corners, there are two cases: (1) A and B are adjacent corners — rotating AB by ±90° gives 2 squares (one on each side of AB); (2) A and B are diagonal corners — the other two corners lie on the perpendicular bisector at distance |AB|/2 from the midpoint, forming exactly 1 square. Total = 2 + 1 = 3 squares.

Solution Steps

  1. Step 1: For A and B on the boundary, the square can vary in size and position, allowing infinitely many squares (similar to circles passing through two points).

  2. Step 2: For A and B as corners, consider two cases: A and B as adjacent corners, and A and B as diagonal (opposite) corners.

  3. Step 3: If A and B are adjacent corners, rotate segment AB by +90° and −90° to find the other two corners. This yields 2 squares (one on each side of line AB).

  4. Step 4: If A and B are diagonal corners, the other two corners must be equidistant from A and B, so they lie on the perpendicular bisector of AB at distance |AB|/2 from the midpoint. These two points together form exactly 1 square.

  5. Step 5: Total = 2 (adjacent case) + 1 (diagonal case) = 3 squares.

PreviousNext