Square - online puzzles

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted

◻

{\displaystyle \square }

ABCD.

Characterizations

A convex quadrilateral is a square if and only if it is any one of the following:

a rectangle with two adjacent equal sides

a rhombus with a right vertex angle

a rhombus with all angles equal

a parallelogram with one right vertex angle and two adjacent equal sides

a quadrilateral with four equal sides and four right angles

a quadrilateral where the diagonals are equal and are the perpendicular bisectors of each other, i.e. a rhombus with equal diagonals

a convex quadrilateral with successive sides a, b, c, d whose area is

A

=

1

2

(

a

2

+

c

2

)

=

1

2

(

b

2

+

d

2

)

.

{\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).}

Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon ( four -sided polygon), and a rectangle (opposite sides equal, right -angles) and therefore has all the properties of all these shapes, namely:

The diagonals of a square bisect each other and meet at 90°

The diagonals of a square bisect its angles.

Opposite sides of a square are both parallel and equal in length.

All four angles of a square are equal. (Each is 360°/4 = 90°, so every angle of a square is a right angle.)

All four sides of a square are equal.

The diagonals of a square are equal.

The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.