Explain how the diagonals of a square perpendicular bisector of each other for
Answers
Step-by-step explanation:
a rectangle with two adjacent equal sides. ... 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.
Internal angle (degrees): 90°
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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).[1] 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 } \square ABCD.
Characterizations
A convex quadrilateral is a square if and only if it is any one of the following:[2][3]
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 {\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).} {\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).}[4]:Corollary 15
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:[5]
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.
A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.
Perimeter and area
The area of a square is the product of the length of its sides.
The perimeter of a square whose four sides have length {\displaystyle \ell } \ell is
{\displaystyle P=4\ell } P=4\ell
and the area A is
{\displaystyle A=\ell ^{2}.} A=\ell ^{2}.
In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.
The area can also be calculated using the diagonal d according to
{\displaystyle A={\frac {d^{2}}{2}}.} A={\frac {d^{2}}{2}}.
In terms of the circumradius R, the area of a square is
{\displaystyle A=2R^{2};} A=2R^{2};
since the area of the circle is {\displaystyle \pi R^{2},} \pi R^{2}, the square fills approximately 0.6366 of its circumscribed circle.
In terms of the inradius r, the area of the square is
{\displaystyle A=4r^{2}.} A=4r^{2}.
Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:
{\displaystyle 16A\leq P^{2}} {\displaystyle 16A\leq P^{2}}
with equality if and only if the quadrilateral is a square.