Squares and RectanglesSquares and Rectangles

A presentation by Ms. A presentation by Ms. Stupp’s favorite students :Stupp’s favorite students :

Juliana BerhaneJuliana BerhaneTiffany JeongTiffany Jeong& Alex Gentile& Alex Gentile

Rectangles

Rectangles are parallelograms with four right angles. In short, they have four perpendicular segments making their sides.A rectangles diagonals bisect each other.A shape is a rectangle if and only if its diagonals are congruent.If a parallelogram has one right angle, then it is a rectangle.

The Golden RectangleThe Golden rectangle is a very

special rectangle. It is a rectangle whose lengths and widths are proportional to the Golden Ratio.The Golden ratio (also known as the divine proportion) is 2 : 1 + radical 5.Throughout the ages, these rectangles have been used in architecture for their pleasing appearance to the eye.

An Oblong

A rectangle that is not a square is colloquially known as an oblong.The term 'oblong' is generally not used today in mathematics. However, it is occasionally used to mean a rectangle whose sides are not equal. In other words, "rectangle" is a generic term for a quadrilateral with four right angles, subdivided into two categories:squares: equal-sided rectangles oblongs: non-equal-sided rectangles The word oblong was once commonly used as an alternate name for a rectangle. In his translation of Euclid’s Elements, Sir Thomas Heath translates the Greek word ετερομηκες [hetero mekes – literally, "different lengths"] in Book One, Definition 22 as oblong. "Of Quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right angled but not equilateral...".

Rectangle (Look at chart)

Each pair of opposite sides are parallelOpposite sides are congruentOpposite angles are congruentDiagonals bisect each otherDiagonals are not perpendicularDiagonals are congruentDiagonals do not bisect two anglesBase angles are congruent

Squares

A square is both a rectangle & a rhombus, therefore it can have two definitions:

A square is an equiangular rhombus.

A square is an equilateral rectangle.

Along with these definitions, a square is a parallelogram as well.

Square Facts

In plane (Euclidean) geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides.The perimeter of a square whose sides have length s is

P = 4s And the area is A = s2

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.Each angle in a square is equal to 90 degrees, or a right angle.The diagonals of a square are equal. Conversely, if the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are about 1.41 times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational.

More Square Facts

If a circle is circumscribed around a square, the area of the circle is about 1.57 times the area of the square. If a circle is inscribed in the square, the area of the circle is about 0.79 times the area of the square. A square has a larger area than any other quadrilateral with the same perimeter.The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry through 90°, 180° and 270°. Its symmetry group is the dihedral group D4. If the area of a given square with side length S is multiplied by the area of a "unit triangle" (an equilateral triangle with side length of 1 unit), which is /4 units squared, the new area is that of the equilateral triangle with side length S.

Squares (Look at chart)

Each pair of opposite sides are parallel Opposite sides are congruent

Opposite angles are congruent

Diagonals bisect each other

Diagonals are perpendicular

Diagonals are congruent

Diagonals bisect two angles

Base angles are congruent

Squares and Rectangles

You see squares and rectangles

everywhere you go . . .

Buildings

Electronics

Spongebob

That’s our show!

Now turn your attention to the overhead and answer our questions!