rectangle

Rectangle. A quadrilateral whose four angles are equal, that is, whose 4 angles are equal to ${90}^{\circ }$.

Any rectangle is a parallelogram.
This follows from angles $\mathrm{\angle }BAD$ and $\mathrm{\angle }ADC$ adding up ${180}^{\circ }$.

Since parallelograms have their opposite sides equal, so do rectangles. In the picture, $AB=CD$ and $BC=DA$.

Rectangles are the only parallelograms to be also cyclic (since opposite angles add up ${180}^{\circ }$.

Notice that every square is also a rectangle, but there are rectangles that are not squares

Rectangles have their two diagonals equal (since triangles $ABC$ and $ABD$ are congruent), A nice result following from this is that the quadrilateral obtained by joining the midpoints of the sides is a rhombus.

Since $PQ$ joins midpoints of sides in triangle $ABC$, we have $PQ=CA/2$. Similarly we have $RS=CA/2$, $QR=BD/2$ and $SP=BD/2$ and thus the sides of quadrilateral $PQRS$ are all equal, in other words, $PQRS$ is a rhombus.

Title	rectangle
Canonical name	Rectangle
Date of creation	2013-03-22 12:02:29
Last modified on	2013-03-22 12:02:29
Owner	drini (3)
Last modified by	drini (3)
Numerical id	6
Author	drini (3)
Entry type	Definition
Classification	msc 51-00
Related topic	Quadrilateral
Related topic	Parallelogram
Related topic	Rhombus
Related topic	ParallelogramLaw
Related topic	Square