QUADRILATERALS

QUADRILATERALS

QUADRILATERALS, known as TETRAGON or QUADRANGLE, is a general term for a four-sided polygon. There are six types of quadrilaterals: square, rectangle, parallelogram, rhombus, trapezoid and trapezium. Each type of quadrilateral has unique properties that make it distinct from the other types.

PARTS OF A QUADRILATERALSIDE is a line segment which joins any two adjacent vertices

INTERIOR ANGLE is the angle formed between two adjacent sides

HEIGHT OR ALTITUDE is the distance between two parallel sides of a quadrilateral

BASE is the side that is perpendicular to the altitude

DIAGONAL is the line segment joining any two non-adjacent vertices

CLASSIFICATION OF QUADRILATERALS

RECTANGLERECTANGLE is essentially a

parallelogram in which the interior angles are all right angles. Since a rectangle is a parallelogram, all of the properties of parallelogram also hold for a rectangle. In addition to these properties, the diagonals of a rectangle are equal. However, the sides are not necessarily equal. A B

d h

C b D

DIAGONALS OF A RECTANGLEA diagonal of a rectangle cuts the rectangle

into two congruent right triangles. Since the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, one can always compute for the third side with the use of the PYTHAGOREAN THEOREM. Thus,

Where d is the diagonal of the rectangleb is the base of the rectangleh is the height or the altitude of the rectangle

PERIMETER AND AREA OF A RECTANGLE

The perimeter is the sum of the four sides. Thus,

The formula to solve the area of the rectangle is given by

SQUAREA SQUARE is a special type of a

rectangle in which all the sides are equal. Since all sides and interior angles are equal, a square is qualified as a regular polygon of four sides.

a

a d a

a

DIAGONAL, PERIMETER AND AREA OF A SQUARE

To find the diagonal, use

To find the perimeter, use

To find the area, use

RHOMBUSA RHOMBUS is a parallelogram in which

all sides are equal. It is also defined as an equilateral parallelogram. The terms “rhomb” or “diamond” are sometimes used instead of rhombus. A rhombus with an interior angle of 45 degrees is sometimes called a “lozenge”.

h b

DIAGONALS OF A RHOMBUSThe diagonals of the rhombus are angle

bisectors of the vertices. By the Cosine law, the diagonals may be obtained in a similar manner like that of parallelogram. Thus,

One can also verify that the angle opposite the shorter diagonal , may be obtained by the formula

PERIMETER OF A RHOMBUS

If b is the measure of one side of a rhombus, then the perimeter is given by

AREA OF A RHOMBUSThe area of a rhombus may be

determined by any of the following ways:

The area is one-half the product of its two diagonals, thus,

The area is also the product of the base and the height, thus

The area is twice the area of one of the two congruent triangles formed by one of its diagonals. Thus,

TRAPEZOID AND TRAPEZIUMA TRAPEZOID is a quadrilateral with one

pair of parallel sides while TRAPEZIUM is a quadrilateral with no parallel sides.

a

h

b

AREA OF A TRAPEZOIDThe area of a trapezoid is equal to the

product of the mean of the bases and the height. In symbols, the are is given by the formula,

Hence, in finding the area of a trapezium, you may use any of the three formulas for the area of a quadrilateral.

PARALLELOGRAMA PARALLELOGRAM is a quadrilateral in

which the opposite sides are parallel.

A B

h

D b C

IMPORTANT PROPERTIES OF PARALLELOGRAM

Opposite sides are equal. Opposite interior angles are congruent. Adjacent angles are supplementary. A diagonal divides the parallelogram into two

congruent triangles. The two diagonals bisect each other.

DIAGONALS OF PARALLELOGRAMIf sides a and b, and the angle are given,

then by the Cosine law, the diagonal may be obtained by the equation:

A B

a d a h

D C b

PERIMETER OF A PARALLELOGRAM

Opposite sides of a parallelogram are equal. Thus, its perimeter is given by the equation,

AREA OF A PARALLELOGRAMThe area of a parallelogram, can be

obtained by any of the following formulas:

where b is the base and h is the height of the parallelogram.

where a and b are the lengths of the sides of the parallelogram and is the interior angle.

NOTE TO ES12KA31. Answer CHAPTER TEST pages 41 to 43 of

“SOLID MENSURATION:UNDERSTANDING THE 3D SPACE” by Richard T. Earnheart, latest edition (green book)

2. Answers only. Use only short bond paper.3. Deadline is until December 7, 2105. You

may submit your assessment (written recitation #1) during our class.

4. Erasures are not allowed.