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Introduction

We have studied many properties of a triangle in previous chapters, and we know that on joining three non-collinear points in pairs, the figure so obtained is a triangle. Now, let us mark four points and see what we obtain on joining them in pairs in some order.

The above figure shows that,

• If all the points are collinear (in the same line), we obtain a line segment. [see Fig. 1]
• If three out of four points are collinear, we get a triangle. [see Fig. 2]
• If all the points are non-collinear, we obtain a closed figure with four sides .Such a figure formed by joining four points in an order is called a quadrilateral. [see Fig 3 & 4]

In this chapter, we will consider only quadrilaterals of the type given in Fig. 3 but not as given in Fig. 4.

A quadrilateral has

• Four vertices
• Four sides
• Four angles
• Two diagonals

Fig. 8.2

136 MATHEMATICS

In quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are

the four vertices and ∠ A, ∠ B, ∠ C and ∠ D are the four angles formed at the

vertices.

Now join the opposite vertices A to C and B to D [see Fig. 8.2 (ii)].

AC and BD are the two diagonals of the quadrilateral ABCD.

In this chapter, we will study more about different types of quadrilaterals, their

properties and especially those of parallelograms.

You may wonder why should we study about quadrilaterals (or parallelograms)

Look around you and you will find so many objects which are of the shape of a

quadrilateral - the floor, walls, ceiling, windows of your classroom, the blackboard,

each face of the duster, each page of your book, the top of your study table etc. Some

of these are given below (see Fig. 8.3).

Fig. 8.3

Although most of the objects we see around are of the shape of special quadrilateral

called rectangle, we shall study more about quadrilaterals and especially parallelograms

because a rectangle is also a parallelogram and all properties of a parallelogram are

true for a rectangle as well.

- - - - - - - - - - - - - - - - - - - - Quadrilaterals - - - - - - - - - - - - - - - - - - - -

• Introduction
• Angle Sum Property of a Quadrilateral
• Types of Quadrilaterals
• Properties of a Parallelogram
• The Mid-point Theorem
• Another Condition for a Quadrilateral to be a Parallelogram
• The Mid-point Theorem

• Exercise 1
• Exercise 2

• Summary

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Divisibility by 2 Rule

Divisibility by 2 Rule 


Almost everyone is familiar with this rule which states that any even numberis divisible by2. Even numbers are multiples of 2. A number is even if ends in 0,2,4,6, or 8. 

Examples of numbers that are even and therefore pass this divisibility test
• 122
• 5670
• 4346
• -2234
• -318
• 31,102

Examples of numbers that are do not pass this divisibility test because they are not even.
• 31
• -103
• 1.50
• 2215

Aristotle of Stagira (384-322 BC)

Aristotle is considered the greatest scientist of the ancient world, and the most influential philosopher and logician ever; he ranks #13 on Michael Hart's list of the Most Influential Persons in History. (His science was a standard curriculum for almost 2000 years, unfortunate since many of his ideas were quite mistaken.) His writings on definitions, axioms and proofs may have influenced Euclid. He was also the first mathematician to write on the subject of infinity. His writings include geometric theorems, some with proofs different from Euclid's or missing from Euclid altogether; one of these (which is seen only in Aristotle's work prior to Apollonius) is that a circle is the locus of points whose distances from two given points are in constant ratio. Even if, as is widely agreed, Aristotle's geometric theorems were not his own work, his status as the most influential logician and philosopher makes him a candidate for the List.

Hippocrates of Chios (ca 470-410 BC)

Hippocrates (no relation to the famous physician) wrote his own Elements more than a century before Euclid. Only fragments survive but it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to have invented the reductio ad absurdem proof method. Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling (the Delian Problem) laid the groundwork for successful efforts by Archytas and others; his circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about "lunes" (certain circle fragments); and some claim Hippocrates was first to trisect the general angle. (Doubling the cube and angle trisection are often called "impossible," but they are impossible only when restricted to collapsing compass and unmarkable straightedge. There are ingenious solutions available with other tools.) Hippocrates also did work in algebra and rudimentary analysis.

Panini (of Shalatula) (ca 520-460 BC)

Panini's great accomplishment was his study of the Sanskrit language, especially in his text Ashtadhyayi. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious elegance that would not be equalled in the West until the 20th century. Linguistics may seem an unlikely qualification for a "great mathematician," but language theory is a field of mathematics. The works of eminent 20th-century linguists and computer scientists like Chomsky, Backus, Post and Church are seen to resemble Panini's work 24 centuries earlier. Panini's systematic study of Sanskrit may have inspired the development of Indian science and algebra. Panini has been called "the Indian Euclid" since the rigor of his grammar is comparable to Euclid's geometry.Although his great texts have been preserved, little else is known about Panini. Some scholars would place his dates a century later than shown here; he may or may not have been the same person as the famous poet Panini. In any case, he was the very last Vedic Sanskrit scholar by definition: his text formed the transition to the Classic Sanskrit period. Panini has been called "one of the most innovative people in the whole development of knowledge."

Pythagoras of Samos (ca 578-505 BC)

Pythagoras, who is sometimes called the "First Philosopher," studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes (from whom Pythagoras acquired a belief in reincarnation); he became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs, so his influence on Plato and Euclid may be enormous. He and his students (the "Pythagoreans") were ascetic mystics for whom mathematics was partly a spiritual tool. (Some occultists treat Pythagoras as a wizard and founding mystic philosopher.) Pythagoras was very interested in astronomy and recognized that the Earth was a globe similar to the other planets. He believed thinking was located in the brain rather than heart. The words "philosophy" and "mathematics" are said to have been coined by Pythagoras.

Despite Pythagoras' historical importance I may have ranked him too high: many results of the Pythagoreans were due to his students; none of their writings survive; and what is known is reported second-hand, and possibly exaggerated, by Plato and others. His students included Hippasus of Metapontum, perhaps the famous physician Alcmaeon, Milo of Croton, and Croton's daughter Theano (who may have been Pythagoras's wife). The term "Pythagorean" was also adopted by many disciples who lived later; these disciples include Philolaus of Croton, the natural philosopher Empedocles, and several other famous Greeks. Pythagoras' successor was apparently Theano herself: the Pythagoreans were one of the few ancient schools to practice gender equality.

Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe." (About the mathematical basis of music, Leibniz later wrote, "Music is the pleasure the human soul experiences from counting without being aware that it is counting." Other mathematicians who investigated the arithmetic of music included Huygens, Euler and Simon Stevin.)

The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first proof. (Apastambha proved it in India at about the same time, and some theorize that Pythagoras journeyed to India and learned of the proof there.) He also discovered the simple parametric form of Pythagorean triplets (xx-yy, 2xy, xx+yy). Other discoveries of the Pythagorean school include the concepts of perfect and amicable numbers, polygonal numbers, golden ratio (attributed to Theano), the five regular solids (attributed to Pythagoras himself), and irrational numbers (attributed to Hippasus). It is said that the discovery of irrational numbers upset the Pythagoreans so much they tossed Hippasus into the ocean! (Another version has Hippasus banished for revealing the secret for constructing the sphere which circumscribes a dodecahedron.)

The famous successors of Thales and Pythagoras included Parmenides of Elea (ca 515-440 BC), Zeno of Elea (see below), Hippocrates of Chios (see below), Plato of Athens (ca 428-348 BC), Theaetetus (ca 414-369 BC), and Archytas (see below). These early Greeks ushered in a Golden Age of Mathematics and Philosophy unequaled in Europe until the Renaissance. The emphasis was on pure, rather than practical, mathematics. Plato (who ranks #40 on Michael Hart's famous list of the Most Influential Persons in History) decreed that his scholars should do geometric construction solely with compass and straight-edge rather than with "carpenter's tools" like rulers and protractors.