A parallelogram whose four sides are equal and whose four angles are right angles is called a square.
Properties of Squares
- Properties of rectangles (equal diagonals)
- Properties of rhombuses (perpendicular diagonals)
- The diagonals intersect perpendicularly at their midpoints and are equal in length.
Organization
Parallelogram ├ Rectangle (angles are 90°) → equal diagonals ├ Rhombus (four equal sides) → perpendicular diagonals └ Square (both a rectangle and a rhombus) → equal and perpendicular diagonals
Example ①
When the length of a square's diagonal is 6√2 cm, find the length of one side.
Solution: The diagonal is bisected perpendicularly, and the triangle is a right isosceles triangle. Side length = 6√2/√2 = 6 cm
When the length of a square's diagonal is 6√2 cm, find the length of one side.
Solution: The diagonal is bisected perpendicularly, and the triangle is a right isosceles triangle. Side length = 6√2/√2 = 6 cm
Key Point
- Square ⊂ rectangle ⊂ parallelogram
- Square ⊂ rhombus ⊂ parallelogram
- Higher-level figures have more properties.
Practice Problem
- Among rectangles, rhombuses, and squares, name all the figures for which “the diagonals are perpendicular” holds.
Answer and Explanation
- Answer: Rhombuses and squares. The diagonals of a rectangle are not necessarily perpendicular (they are equal, but may not be perpendicular).